mental
Physics
7c;LjiKw;;
E. SEGRE
Editor
.
EXPERIMENTAL
NUCLEAR PHYSICS
CONTRIBUTORS
VOLUME I
Julius Ashkin Kenneth T. Bainbridge
Hans A. Bethe Norman F. Ramsey
Hans H. Staub
VOLUME II
Philip Morrison Bernard T. Feld
VOLUME III
In preparation
EXPERIMENTAL
NUCLEAR PHYSICS
VOLUME II
E. SEGRE, Editor
P. Morrison and B. T. Feld
JOHN WILEY & SONS, INC., NEW YORK
CHAPMAN & HALL, LIMITED, LONDON
few. i £i>/
CLASSICS b,^C NO.
, ■„JjJ4^Ss _
Checked I ^te*V
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Copyright, 1953
BY
John Wiley & Sons, Inc.
All Rights Reserved
This book or any part thereof must not
be reproduced in any form without
the written permission of the publisher.
Library of Congress Catalog Card Number: 525852
PRINTED IN THE UNITED STATES OP AMERICA
PREFACE
At the end of World War II many physicists who had been mobilized
for the war effort returned to university work and to pure research; a
great number of them had worked on nuclear problems and were anxious
to resume investigations in this field. Moreover there was a large influx
of students eager to start nuclear investigations.
The need was keenly felt for a book which would bring the experimen
talist up to date in experimental techniques, point out to him the sig
nificant facts and data, and indicate the broad lines of theoretical inter
pretation.
It was immediately apparent that the field of nuclear physics had
grown so much and the various branches had become so specialized that
no one person could hope to write a book like the famous treatises of
Rutherford (which, however, because of the evolution mentioned above,
had by 1930 already become Rutherford, Chadwick, and Ellis), Curie,
and Kohlrausch. A cooperative effort like the GeigerScheel Handbuch
der Physik seemed the only solution. Individual authors could undertake
to prepare reasonably complete treatises on a restricted field in which
they are quite authoritative. By keeping the discussions relatively
short, it became possible for a group of authors to cooperate without cur
tailing their research activity. An incentive for several of the authors,
indeed, was the desire to read the contributions of the others.
This work is the outcome of that effort. We hope that it will be useful
to the serious student and to the research worker in the field. Each part,
with its bibliography, should be sufficient to inform the reader about
the main results obtained in nuclear physics up to the end of 1952 and
enable him to go directly to the original literature or to the several
excellent collections of periodical reviews which are currently appear
ing (Annual Reviews of Nuclear Science, Reviews of Modern Physics,
Ergebnisse der exakten Naturwissenschaften, Progress in Nuclear Physics,
and others) for further details.
E. Segre
Berkeley, California
June, 1958
CONTENTS
Part VI. A Survey of Nuclear Reactions
Philip Morrison
1 . The Conservation Laws 1
A. Application of EnergyMomentum Conservation. B. Conservation of
Angular Momenta.
2. The Data of Nuclear Reactions 15
A. Yields and Cross Sections. B. The Measurement of Yields and Cross
Sections. C. Types of Reactions: A Guide to the Literature.
3. The Nuclear Model 25
A. Qualitative Account of the Model. B. Calculation of Level Densities.
4. Nuclear Level Widths 45
A. Level Widths and Reaction Cross Sections: Statistical Relations.
B. Calculation of Level Widths.
5. The Course of Nuclear Reactions 54
A. The Steps of the Reaction. B. The Contact Cross Section. C. The
Disintegration of Compound States.
6. The Dispersion Theory: Resonance Reactions 64
A. The OneBody Model and Its Difficulties. B. The Dispersion Theory
for an Isolated Resonance. C. The Generalized Theory of Dispersion:
Many Levels and Many Decay Modes. D. Statistical Estimates.
7. Some Typical Nuclear Reactions 83
A. Resonance: The Region of Dispersion Theory. B. Reactions without
Marked Resonance.
8. The Deuteron as a Projectile 110
A. The OppenheimerPhillips Reaction: LowEnergy Stripping. B. Strip
ping Reactions at Higher Energy.
9. Radiative Processes in Nuclear Reactions 114
A. The Multipole Classification. B. Calculation of Radiation Widths.
C. PhotoInduced Reactions.
10. Nuclear Fission 123
A. The Energetics of Fission. B. The Products of Fission. C. Fission
Cross Sections.
11. Nuclear Reactions at High Energy 141
A. The Nuclear Cascade. B. Correlations among Nucleons. C. The Op
tical Model for the Scattering of Nucleons. D. The Processes of Nuclear
Deexcitation at High Energy. E. Mesons: Virtual and Real.
vii
viii Contents
Part VII. The Neutron
Bernard T. Feld
1. Properties and Fundamental Interactions 209
A. Discovery. B. Properties. C. Fundamental Interactions.
2. Interaction with Nuclei 247
A. Introduction. B. General Considerations. C. Types of Neutron Re
actions.
3. Sources and Detectors: Neutron Spectroscopy 357
A. Introduction. B. Neutron Sources. C. Neutron Detectors. D. Slow
Neutron Spectroscopy. E. The Calibration of Neutron Sources.
4. The Interaction of Neutrons with Matter in Bulk 460
A. Introduction. B. Diffusion of Monoenergetic Neutrons. C. The
SlowingDown Process. D. The Nuclear Chain Reaction.
5. Coherent Scattering Phenomena with Slow Neutrons 508
A. Introduction. B. Neutron Diffraction and the Structure of Matter.
C. Neutron Diffraction and the Determination of Nuclear Scattering
Amplitudes. D. Magnetic Scattering and Neutron Polarization.
Author Index 587
Subject Index 595
PART VI
A Survey of Nuclear Reactions
P. MORRISON
Cornell University
The study of nuclear reactions parallels the familiar study of chemical
reactions. Most of the root ideas of chemistry are fundamental for the
nuclear physicist. The equation of the reaction, the heat of reaction,
the rate of reaction, the "balancing" of the equation — all these have
their nuclear counterparts. The fact that the characteristic energy
release is the millions of electron volts associated with the binding of
nuclear particles, rather than the few volts which result from molecular
binding, is the most striking inherent difference between the two fields.
The nuclear physicist can detect the reaction not of moles or micromoles
of his reactants, but of individual 'particles. This has meant that from
the beginning the nuclear physicist has studied not equilibria and the
laws of mass action, but the properties of single collisions. Just as the
chemist has recently come to regard the individual collision as the center
of attention for really fundamental understanding of reactions, so the
physicist has now come to study largescale reactions and even thermal
equilibria in the nuclear domain in the light of his newer interest in
astrophysical and chain reaction problems. Although the two points
of view have tended to merge, it is our task to survey nuclear reactions
primarily from the point of view of understanding individual collisions.
Clearly this is the key to any genuine insight.
SECTION 1. THE CONSERVATION LAWS
Nuclear reactions, like chemical ones, can be more or less complex.
The typical reaction — with which we shall be most concerned — involves
a stationary or "target" nucleus bombarded by a relatively light incident
nuclear projectile. The products of the reaction may be one, two, or
even more nuclei. In the nuclear reactions of most interest before the
development of the very highenergy accelerators in 1947, one to three
products were by far most common. The target nucleus changes into a
nuclear species differing by only a few units in A, the mass number or
1
2 A Survey of Nuclear Reactions [Pt. VI
number of nucleons (protons and neutrons) in the nucleus, and in Z, the
number of nuclear protons and, hence, the number of electrons in the
neutral atom. Such reactions are designated either in the somewhat
redundant notation taken from the chemists, an example of which might
be uNa 23 + 1H 1 — * i2Mg 23 + on 1 [on 1 = neutron], or in an abbreviated
scheme, due to Bothe. In the latter, one writes first the target nucleus,
indicating its mass number and chemical character (often giving the
nuclear charge Z as well), then in parentheses the symbols for the bom
barding and emitted particles in order, and finally the residual nucleus.
The above reaction would be written Na 23 (p,n)Mg 23 . If there are
more than two products of reaction, one might write, for example,
C 12 (p,pn)C n for the reaction 6 C 12 + 1H 1 » 6 C n + {H. 1 + on 1 .
What physical properties of the system remain unchanged throughout
the reaction? These conserved properties provide very valuable infor
mation for every reaction. We list conserved properties :
1. In no observed process does the total electric charge change. A
proton can change into a neutron, but a positron or positive meson must
appear, or an electron disappear, in the process. The familiar creation
of electronpositron pairs is a dramatic confirmation of this principle.
2. To this date (1952) no process has been observed in which the
total number of nucleons, i.e., the total value of A, is different on the
two sides of the reaction. Presumably this is not a fundamental property
of nucleons; the creation of positive and negative "proton" pairs is
expected when sufficient energy is available. No such process has yet
been observed. Nucleon number is conserved in all known reactions.
3. The constants of motion of classical mechanics are conserved, at
least to an order of accuracy beyond experimental interest. These are
total energy, momentum, and angular momentum. The energies of
chemical binding, and the accompanying forces and torques, are negligi
ble compared to the energy transfers in typical reactions; so the colliding
particles can be regarded as a mechanically isolated system. (For some
special reactions involving very slow neutrons, the chemical binding and
the thermal motion of the target nuclei must be considered. Even the
whole of a macroscopic crystal may be taken as the target in some situa
tions. Compare Part VII, Section 5.) The conservation of total
energy must of course include the energy equivalent of mass changes
of the reactants, for this will generally be a large fraction of the available
kinetic energy. The conservation of momentum guarantees that the
incident and product nuclei move in a plane if only two product nuclei
result. The total angular momentum includes of course the intrinsic
or spin angular momentum of the reactants combined with the orbital
Sec. 1A] The Conservation Laws 3
angular momentum of their relative motion. It is this total which must
be conserved, though the breakdown of the total into intrinsic and orbital
angular momenta may not be predicted in general. The experimental
determination of the angular momentum of individual particles is nearly
out of the question, but the statistical angular distribution of reaction
products depends on the angular momentum relations. Measurements
of such distributions are important aids to analysis of nuclear reactions.
4. The "constants of motion" peculiar to quantum mechanics are
also conserved. These are two in number: the parity and the statistics
(S3). 1 The parity refers of course to the behavior of the wave function
of the system upon inversion of all coordinates in the origin, i.e., upon
the changing of right to lefthanded axis systems. No physical property
of an isolated system can depend upon the kind of axes used, but the
wave function can change sign under such a transformation without
affecting any expectation values. Whether the wave function changes
sign or remains the same under such an axis inversion is indicated by
the parity: even for wave functions remaining unchanged, odd for those
changing sign. The parity of the initial wave function of the whole
system must be conserved.
The statistics of the system is related to these constants. The inter
change of all coordinates of identical particles in any system can lead
only to change of sign or to an unchanged wave function. All particles
must belong either to the class whose wave functions change sign when
identical particles are interchanged (these are called FermiDirac
particles) or to the class whose wave functions must remain unchanged
(these are called BoseEinstein particles). From the relativistic theory
of wave fields, it can be shown that particles with halfinteger intrinsic
angular momentum are Fermi particles; those with integer spin, Bose
particles. In ordinary reactions, where the nucleons are simply rear
ranged, this conservation law leads to no new conclusions; but in proc
esses where new types of particles — mesons, neutrinos — are created or
destroyed, this condition sharply limits the processes which can take
place.
A. Application of EnergyMomentum Conservation
1. Elastic Collisions. The simplest application of these principles is
to the elastic collision of two free particles. The slowing down of neu
trons by elastic collision is the bestknown example (see Part VII,
Section 4), but, because the treatment is the basis for more complicated
processes, we shall give the theory here, using the less familiar, but
1 References for the text of Part VI will be found on pages 189 to 192.
A Survey of Nuclear Reactions
[Pt. VI
increasingly useful, relativistic form (B7). The nonrelativistic formulas
are of course contained in the lowenergy limit.
We begin with the target particle at rest in the laboratory coordinate
system C. It is a simplification to make a transformation to a moving
system C in which the laboratory and target particle are of course mov
ing, but the total momentum of the colliding particles is zero. Lorentz
transformation from this system to C will give us the results we want;
we know that the energymomentum conservation is automatically
satisfied in any Lorentz frame if it holds in C".
Let the target particle of rest mass m 2 be stationary in C, and the
other particle of mass mi be incident with velocity v along the x axis.
In the system C" the particles approach each other with velocities «/
and u 2 respectively. We require the particles to have the equal and
oppositely directed momenta p' before collision. Then
V =
miUi
m 2 u 2
(1fr' 2 )^ (1/J 2 ' 2 ) M
where j3' = u'/c. Using the notation 7' == 1/(1 — /3' 2 ) M , we have
(la)
Squaring,
c
«i 8 W 2  1) = m 2 2 (y 2 ' 2  1)
(lb)
(2)
Since the total momentum remains zero after the collision, the final
momenta are easily seen to be equal and opposite, but the particles
now are receding instead of approaching. Compare Fig. 1. If particle 1
Fig. 1. Momentum relations for collision in centerofmass system.
is scattered through the angle 6' in C", particle 2 will come off at an angle
w — 6' with respect to the initial direction of particle 1. The velocities
are not changed because the kinetic energies are conserved in an elastic
collision.
Sec. 1A] The Conservation Laws 5
We now make a simple Lorentz transformation to the laboratory rest
system C. If particle 2 is initially at rest in C, C must be moving relative
to C with velocity u 2 in the direction of the incident motion. Now we
know that a fourvector is formed by the momentum components and
the total energy E/c 2 , transforming under Lorentz transformation like
the coordinates x, y, z, t. Therefore the Lorentz transformation gives
for the initial momentum of the first particle in the system C" (omitting
a factor of m\C throughout) :
(7/ 2  1) K = 7 2 '[(7 2  1) K ~ /Vy] = 7 2 '(7 2  1)*  (7 2 ' 2  1) M 7
(3)
where m,ic(y 2 — 1)^ is the momentum, and miC 2 y the total energy of
particle 1 in the laboratory system C. The kinetic energy is m\C 2 (y — 1).
Using (2) and (3), we find
■ 2/
Ti = 7 : , .^u (4a)
72 =
1 +
(4b)
To obtain relations between the scattering angles, we again use the
fact that the momentum and energy E, divided by c 2 , form a fourvector.
The total energy E 2 of particle 2 after the collision and its momentum
components, in the laboratory system, are given by
E 2 = 72 W  c/SaV cos 6') (5a)
Px 2 = 7 2 y (p' cos 0' +  /9 2 'tf 2 'j (5b)
P»2 = P'sinfl' (5c)
where the incident direction is the x axis, and all the momenta lie in a
plane. The energy of particle 2 in C is E 2 ' = m 2 c 2 y 2 ', and its momen
tum p' = m 2 y 2 'c(3 2 ' Dividing (5b) by (5c), we find the angle of scatter
ing $2 of particle 2 in the laboratory system C:
1 1
tan 9 2 = cot  6' (6)
72' 2
6 ' A Survey of Nuclear Reactions [Pt. VI
The minus sign means that, if particle 1 is scattered above the x axis
into the region of positive y, particle 2 recoils below the axis. If 0' is
180° in the centerofmass system C, 6 2 = and particle 2 recoils for
ward with maximum energy. As 0' decreases toward zero, particle 2
recoils with decreasing kinetic energy at angles approaching 90°. The
case 0' = corresponds to grazing collision, in which particle 1 loses no
energy and particle 2 recoils with zero velocity at 90°.
The recoil kinetic energy of particle 2 in the laboratory system is
7' 2 = E 2 — m 2 c 2 . From (5a)
T 2 = m 2 c 2 (y 2 ' 2  1)(1  cos d') (7a)
Using (6) and (4b), we can write this in terms of the angle d 2 . Finally
we may introduce the total energy E = m,\<?y and momentum p =
mic(y 2 — 1)^ of the incident particle, getting
p 2 COS 2 2
T 2 = 2m 2 c 2 (7b)
(E/c + m 2 c) J — p 2 cos 2 2
Evidently 2' 2 (max) = 7' 2 (0 2 = 0).
It is of interest to compute the maximum fraction of the kinetic
energy of the incident particle which can be transferred. T = E — m\C 2
= mic 2 (T — 1), and
7V max) _ 2( 7 + \)m x m 2 _ 2( 7 + \)m x m 2
T m 2 + 2ymxm 2 + m 2 2 2(7 + l)mim 2 + {mi — m 2 ) 2
This fraction increases monotonically with incident energy from the
value familiar in the nonrelativistic limit, 4m 1 m 2 /(mi + m 2 ) 2 , to unity
for very high energies, with 7 » 1. If the particles have equal mass,
the maximum recoil energy is of course always equal to the incident
energy.
If the particle incident is light (e.g., electron or photon), particle 2
can acquire relativistic energies only if p ■= m 2 c 2 or 7 = m 2 /m\. This
is also the condition for transfer of a large fraction of energy from par
ticle 1 to particle 2, if a heavy particle is incident on a light target particle.
We now discuss the relation between the scattering angle 0i of particle
1 and its energy loss. The total energy of particle 1 in the laboratory
system C corresponding to a scattering through 0' in C is
E x = m lC 2 [7i'72' + (7i' a  1) } ^(72' 2  1) H cos 0'] (9)
where the 7"s are as given by (4a) and (4b). Now we need cos 0' in
terms of the angle measured in the laboratory system. For this we
Sec. 1A] The Conservation Laws 7
apply the Lorentz transformation to the energy and momentum of
particle 1 in the manner of (3). This shows
1 sin 0'
tan 0! = (10)
y 2 (cos 0' + TO 1 Y 1 '/W12Y2')
Solving for cos 0' in terms of tan 0\, we get a quadratic equation with
two roots, indicated by the ± sign:
cos 0' =
mi ,
7l ' T2 ' tan' 5 0x
(1 + T2' 2 tan 2 X ) L m 2
2 ™ 2
1 5 tan 2 ^)
(11)
Since the energy for a definite scattering angle in the laboratory system
is given by (9) in terms of cos 0', it appears that there are two possible
values for the energy of a particle elastically scattered through a definite
angle. It is instructive to examine the situation graphically.
For nonrelativistic velocities (7 = 1), Eq. (11) gives the tangent
of the scattering angle in C in terms of functions of the angle 0' in C".
For higher velocities, we can define an auxiliary angle ^ by the relation
tan \p = y 2 tan0j. This angle \f/ can be obtained geometrically by
adding a vector of length m^ix /m 2 y 2 along the x axis to a unit vector
whose components are cos 0' and sin 0'. The center of a unit circle
represents C", and the scattered particle may go off in any direction,
along a unit radius vector. The auxiliary angle ^ is then just the angle
between the x axis and the resultant of the radius vector in C added to
the vector miyi'/m 2 y 2 '. The construction for the three cases we must
consider is shown in Fig. 2.
In case (a), with mi > m 2 , there are two possible directions of scatter
ing in C" (i.e., two distinct orientations of the unit vector) for one scatter
ing angle 0i in C. There are thus two values for the energy of particle 1 ,
in agreement with (11).
For the case mi < m 2 , Fig. 2b is appropriate. Here the a>axis vector
is less than unity, and the construction gives only a single angle 0' to
be associated with a definite i/'. The analogue to the second orientation
of the unit vector in case (a) corresponds to the angle (x — \f/) and gives
no additional solution. One of the roots of (11) is to be rejected if
mi < m 2 . From the diagram we can see that, if 0i is less than 90°, we
should take the more positive value of cos 0' in (11); for X greater than
90°, the more negative one.
8
A Survey of Nuclear Reactions
[Pt. VI
We shall treat case (c), mi = m 2 , as a limit of either (a) or (b). From
the figure we can see that there is an upper limit to ^ and thus to the
angle 1; corresponding to the case where the line defining \p is tangent
(a) m 1 >m z
Unit circle
(b) m 1 <m 2
(c) m^Mj
Fig. 2. Scattering angle relations for elastic collisions. Incident particle is mi;
target particle, m^. The angle 41 determines the laboratory scattering angle. See
text for construction.
to the unit circle, and angles 0'(u, 0\ 2 ) have coalesced to a single
value. The radical in (11) must vanish at this maximum value:
tan^""" =
m 2
m\ — m 2
(12)
9i (max) = arc s i n
m 2
nil
In case (b), with m\ < m 2 , scattering can clearly occur for all angles
between and t. In the limiting case (c), mi = m 2 , the condition of
tangency is reached for ir/2.
In the special case (c), all the formulas above are much simplified.
From (4a) and (4b) we get
Ti' = 72' = {—£) ( 13 )
Sec. 1A] The Conservation Laws 9
If T* is the kinetic energy of the particle scattered through angle 6 in C,
either (5a) or (7b) gives
2(7  1) cos 2
(7 + 1) ~ (7  1) cos 2
and, with T = mc 2 (y — 1),
T cos 2 e
T* = mc 2 — (13a)
1 + f (7Vmc 2 ) sin 2
mi — m 2 = m (13b)
Although there are still two roots of (11), one of them always corresponds
to 0' = 180°. This means of course that one of the colliding particles
is projected in the forward direction with the full incident energy, while
the other remains at rest. The second root of (11) gives (13a).
The familiar nonrelativistic form of (11) is easy to obtain. If
»ti = m 2 , the scattered particles always come off at right angles to each
other, with energies proportional to the squares of the cosines of the
angles of deflection. For higher velocities, however, the angle included
between the directions of the outgoing particles is always less than a
right angle. Changing the sign of 2 in (6) and using (10) and (13a)
and (13b), we find
2
tan di tan 2 = (14)
1 + 7
Since y is always greater than unity, 0i + 2 is less than 90° except for
the grazing collisions in which particle 2 may be thought of as scattered
at 90° with zero velocity. A sufficiently accurate measurement of
angles 0i and 2 would serve to determine the incident energy. This
method is more sensitive for large values of y.
Finally we set out the familiar relations for the nonrelativistic case.
If T 2 is the kinetic energy of recoil of particle 2 and T is the incident
kinetic energy of particle 1,
T 2 = ^cos 2 d 2 T (15)
(mi + m 2 ) i
For the final kinetic energy of particle 1, from the proper limit in (9),
we obtain
\mi + m 2 / L \m\/
/m 2 2 . y
± 2 cos 0i I — r — sin 2 0! I
The treatment of the two roots is the same as in Fig. 2.
(16)
10 A Survey of Nuclear Reactions [Pt. VI
2. Collisions with Creation of New Particles. For sufficiently high
energy collisions of nuclear particles, not only are rearrangements of
nucleons possible, in which kinetic energy is transformed into binding
energy (or the reverse), but also new particles may be produced whose
rest energy comes from the kinetic energy of collision. In a collision in
which additional particles are produced whose rest energy totals Mc 2 , it
is clear that energy conservation requires that at least the incident kinetic
energy exceed Mc 2 . In addition, because the conservation of momentum
requires that the center of mass of the whole system move with un
changed momentum, we must usually provide additional energy to fulfill
this condition.
Let us consider the threshold value of the kinetic energy of the incident
particle, ra x , on a stationary target particle m 2 (F8). The threshold
is the minimum energy below which the production cannot take place.
How probable the production becomes just above threshold is of course
impossible to tell in general; it will depend upon the particular inter
actions being considered. It is easy to see that at the energy of threshold
the total energy of all particles is a minimum after collision. In the
frame C" we know that the final momentum must be zero. These two
conditions can be satisfied if all particles after collision are individually
at rest in the centerofmass system C". Then the initial kinetic energy
must have been just equal to the increment in rest mass energy, Mc 2 .
The total energy in the C" system before collision is
E t ' + E 2 ' = mi c 2 yi ' + m 2 c 2 y 2 '
and from (4) we can write this
Ex' + E 2 ' = c 2 (w! 2 + 2ym } m 2 + m. 2 2 )' A (17)
Equating it to the energy after collision,
Ei' + E 2 ' = c 2 ( mi + m 2 + M)
Solving (17) for y, we obtain the threshold kinetic energy T of mi in the
laboratory system from the relation T = m A c 2 (y — 1). This gives
Mc 2 {m l + m 2 + IM)
T = LJ *—L (18)
m 2
To create a nucleon pair from a protonproton collision, for example,
would require T at least 2 ■ m p c 2 (3m p /m p ) = 6m p c 2 . If the pair were
made by a gammaray or electron incident {m x ^ 0), the threshold
would be T = 4m p c 2 .
In such collisions the target particle need not be at rest. Here we
may think of a target nucleon as one of the nucleons bound in a nucleus,
Sec. 1A] The Conservation Laws 11
and thus moving with the velocity characteristic of its zero point vibra
tions. If the collision occurs when the target nucleon is moving with a
velocity component directed toward the incident particle, less incident
beam energy will be needed. If the target particle m 2 has the initial
velocity v 2 in the x direction, Lorentz transformation shows that the
threshold energy of (18) becomes reduced to the value
T min = y*(T  v 2 p) + m lC 2 ( T2  1) (19a)
where p is the initial momentum of particle mi, and 72 = (1 — v 2 2 / c 2 )~ V2 .
A further reduction comes about if the collision occurs with capture of
the incident particle so that binding energy can be released.
It is of some interest to discuss the possible angular and energy dis
tribution of the produced particles. If we restrict ourselves to the case
in which the target particle is at rest, and in which only one new particle
of mass M is made, we can draw some simple conclusions. It is clear
that at the threshold energy there is only one final condition: all the
particles move forward with the same velocity, since they are all at rest
in the centerofmass system. As the energy exceeds the threshold, the
created particle will in general be able to travel in a distribution of
angles around the forward direction. The details will of course depend
on the nature of the interaction. But some limits can be given in general.
It is clear that the maximum momentum of the new particle will corre
spond to a case in which M is traveling forward and the two original
particles move opposite to M in the centerofmass system, with the
same speed for each. This leads to a value for the maximum kinetic
energy of M in the laboratory system :
Mc 2 + r m « =
\c 2 [{ynii + m 2 )A + mi(y 2  \) }i B]/{mi 2 + 2ym 1 m 2 + m 2 2 ) (19b)
with A = M 2 + 2(t  \)m x m 2 and B = {[2(7  l)m 1 m 2  M 2 ] 2 
AM 2 (nix + %) 2 ! ^. This has a very simple form at the threshold:
Mc 2 Mini! + M/2)
T max = — (20)
m 2 mi + m 2 + M
The angular distributions will be controlled by the relation between
the maximum velocity with which M can move off in the centerofmass
system and the velocity of C with respect to C". Arguments similar to
those accompanying Fig. 2 show that here too there are two possibilities :
the new particle will be emitted only in part of the forward hemisphere
if its maximum centerofmass velocity vm = c 2 pm'/Em' is less than
the velocity of the centerofmass system, c/3 2 '. Then the situation is
12
A Survey of Nuclear Reactions
[Pt. VI
like Fig. 2a, and a minimum energy for M in the laboratory exists; it is
given by (19) with a minus sign instead of a plus sign before the radical.
If v M ' > c/V, however, the case is like that, of Fig. 2b, and all angles of
emission are allowed in the laboratory frame. The minimum energy for
M is then zero in the laboratory system.
3. Reaction Energies. The most studied nuclear reactions up to the
present time involve neither elastic collisions nor the production of new
particles. They consist of the rearrangement of nucleons, with the
ejection of different nuclei having correspondingly changed binding
energies and rest masses. Just as in chemistry, the heat of reaction Q is
a significant quantity. The energy Q is the energy released as kinetic
energy (or the energy of photons, etc., set free) at the expense of the
internal energy of the colliding systems. We may define Q by either
of the relations:
Q = c 2 ( £ tru EmA = I Ti + £ T f (21)
V initial final / initial final
where the notation is that of Section 1A2, and the equivalence of the
definitions follows from the conservation of total energy; T { is defined
as kinetic energy. We consider first the case of a typical reaction of the
twoproduct type: T(i,p)R.
It is very easy to treat this case with the target nucleus at rest. Then
conservation of momentum in the laboratory system yields the vector
relation P; = p p + Pr. We have only to square this, employ the
relativistic connection between momentum and kinetic energy T,
p 2 = T 2 /c 2 + 2mT, and substitute in the definition for Q, eliminating,
say, T R . We have
\ 2 T
Q = T P Ti + m R c 2
+
m R / m,
X
T T
IrripC ■ miC
/m p V T p / [ T p \
\tor/ m p c 2 \ m p c 2 /
(— +2)
{ C 2 \TOjC 2 /
m p &/ \ rriicr/
m p c
2m p nii
ra R 2
 1
(22a)
This may be written in the lowenergy limit by expanding the radical,
neglecting T/mc 2 compared to unity for all particles. The familiar
result follows:
m / m v\ ( w, \ (m v T v mi'.
Q = T p [l+—)Ti[l )2 P P
\ TOr/ \ Tor/ Tor
miTi)>
Vz
cos 6 P (22b)
Sec. IB] The Conservation Laws 13
In reactions of this type, with only two product particles, there is evi
dently a unique value of T p for a given recoil direction and a given inci
dent energy. When the reaction produces three or more particles, no
such simple connection exists. The case is then parallel to the discussion
in Section 1A2. A distribution of energies results. If the reaction takes
• place in two independent stages, first the emission of two particles, then
the breakup of one of the pair, somewhat simpler relations can be
obtained. Thus, study of the energy distribution can give some insight
into the mechanism of the reaction.
The rigid correlation between energy of outgoing particle and angle of
emission has been much exploited as a laboratory means for obtaining
particles of a welldefined energy. For neutrons especially such mono
energetic beams can hardly be obtained in any other way. Simply as
an example, it is interesting to note that the very exothermic reaction
T 3 (d,w)He 4 used with a wellcontrolled 1Mev beam of deuterons pro
duces neutrons ranging from almost 16 Mev to under 13 Mev in the
backwards direction. A study of a set of such reactions has been made
which makes possible the production of monoenergetic neutron beams of
energies from a few kilovolts up to about 20 Mev; a source of charged
particles of welldefined but rather modest energies of a couple of Mev
was used. A valuable review by Hanson et al. (H2) fully discusses the
several reactions used most frequently and gives a graphical treatment
of the fundamental relations. Graphical treatments of the relativistic
case, very useful when it is desired to reduce experimental data, have
been given in (B16) and especially in (Ml).
It is worth while to make the remark in closing that, in all the relati
vistic formulas above, the case in which some particle (i) is a photon can
be obtained by replacing the total energy Ei = mic 2 y by the quantum
energy hv, and neglecting m, wherever it occurs without a factor 7.
B. Conservation of Angular Momenta
The complementary relation in quantum mechanics between energy
momentum and spacetime leads to straightforward experimental use
of the ideas of energy and momentum, as we have seen. The important
canonical variable, angular momentum, is conjugate to angle. We
measure the angular momenta of quantum systems mainly by obtaining
statistical distributions in angle. The typical nuclear reaction is carried
out with a beam of particles ; the very term beam implies a more or less
sharp definition of the direction of motion, and hence a necessarily
rough limitation on angular momentum. An infinite plane wave, indeed,
contains all angular momenta of orbital motion, with definite ampli
tudes, and in addition requires supplementing by the wave function
14 A Survey of Nuclear Reactions [Pt. VI
factor representing the intrinsic angular momentum of the particles in
the beam. Only when there is some reason to limit the states of angular
momentum which are of interest — either simply formally in some
systematic counting of all angular momentum states (as in the method
of partial waves) or physically because for some reason only a limited
number of angular momenta contribute to a given reaction — can the
angular momentum conservation law be of much value. For example,
in the familiar case of thermal neutron reactions, where the incident
wavelength is large compared to the region of possible interaction, only
the spherically symmetric component — angular momentum equal to
zero — of the incident wave can possibly contribute. Other parts of the
wave are of vanishingly small amplitude; the "centrifugal barrier"
keeps them from the region of interaction. If any specification can be
made of the angular momentum (hence of the variation with direction)
of the incoming wave, general rules may often become helpful.
The total angular momentum is certain to be conserved; the forces
between nucleons are, however, noncentral, so that there is a tendency
to interconvert intrinsic or spin angular momentum with that associated
with orbital motion. Only in special cases can the relative apportioning
and hence the complete angular distribution be obtained (Y2). The
spin angular momentum affects, of course, not the direction of motion
in space but the orientation of the spin axis of the particle concerned.
If an unknown spin change can occur, this will clearly affect predictions
of angular distribution in a way calculable only under further specifica
tion of the interactions involved. There are three general results, inter
esting to present, which apply quite generally to all nuclear reactions
between unpolarized target and unpolarized incident particles:
1. As is clear on physical grounds, there will be axial symmetry about
the direction of the incident beam.
2. If, among the incoming partial waves, only those of angular mo
mentum L or below contribute appreciably to a reaction, the angular
distribution of any single product particle cannot be more complicated
than that of the incoming contributing partial wave. Indeed, the angu
lar distribution of the outgoing particle will be a polynomial in cos 6,
where 6 is the polar angle of emission relative to the beam direction, of
degree no higher than 2L. This holds independently of the spin of any
particles or of the number which take part. It is restricted to non
relativistic velocities for the incoming particle. In relativistic cases,
one higher power of L may occur, but in general it will be reduced by a
factor of the order of (v/c). If several partial waves of different L con
tribute, the polynomial will generally contain all powers of cos 6 from
Sec. 2A] The Data of Nuclear Reactions 15
the maximum down; if only one wave contributes, only even powers of
cos 6 can occur. These results refer, of course, to the centerofmass
frame of reference.
3. If the distribution of product particles contains any odd powers in
cos 8, i.e., if it is not symmetrical with respect to the plane normal to the
beam, two (or more) intermediate states are involved with opposite
parities. Thus an even polynomial in cos 9 is a typical consequence of
reactions involving a marked resonance; this will be discussed more
fully later.
Most frequently such considerations yield useful information only
when there is additional information, for example data permitting the
original specification of what partial waves can contribute. This implies
some statements about the mechanism of the reaction, some definite
nuclear model, and some assumption about the character of nuclear
states and of interactions. Then the selection rules which result are
often powerful discriminants between possible alternatives. Such cases
occur most frequently in nuclei lighter than neon, in which region some
progress has been made in constructing detailed models for nuclear
systems. Examples will be given in the proper places later.
SECTION 2. THE DATA OF NUCLEAR REACTIONS
The conservation laws are satisfied in every single nuclear collision.
But, besides data on the constants of motion, nuclear physics also is
concerned with knowing how many nuclear processes of a definite kind
take place under given conditions. It is evidently beyond us to say
whether or not the next proton, say, will initiate a transmutation in a
given fluorine nucleus, but we can obtain from both experiment and
theory the probability of the process. We can expect to predict with
calculable accuracy the fraction of the beam of incident protons which
enters the target nuclei.
A. Yields and Cross Sections
The simplest expression for the probability of a nuclear reaction is
the yield. This is frequently but a semiempirical expression, stipulating,
for a particular experimental arrangement, how many processes occur
per incident particle. The statement that a yield of 1/2000 was obtained
for the reaction Li 7 (p,a)a with a thick target at 3 Mev means only that
one lithium nucleus divides for every 2000 protons stopped in the target
material. The specification is obviously incomplete. For data about
the probability of the reaction itself one would need to know how many
16 A Survey of Nuclear Reactions [Pt. VI
encounters occurred and at what energies, whereas here the protons lose
energy as they penetrate the target, which is often not even a single
nuclear species but a mixture of isotopes or even a compound. Improve
ment could be obtained by using a target so thin that the energy loss of
the beam protons in traversing it was negligible ; here at least the energy
would be sharply defined. Thintarget or thicktarget yield data are
common in the literature; each requires special interpretation before
any absolute number can be obtained (CO, Rl, R2). Such a form of
presenting data implies that only some relative feature of the reaction
is under study.
The familiar cross section is a complete specification of the probability
of nuclear reaction. A cross section oab for a reaction in which the par
ticle A is incident with specified properties and the specified set of
particles B emerges can be defined by the expression n# = Ia^abN.
Here N is the number of target particles presented to the beam (for
example, for thin targets the product of beam area, target thickness, and
number of target atoms per cubic centimeter gives JV), n# is the number
of the specified particles B emitted per second, and I a is the incident
flux of particles A, the number per unit area and time. 1
A cross section may be specified for any process or partial process:
for example, we can assign a cross section for the emission of gammarays
of energy E in the direction specified by the polar angles (8, <p) when
protons of energy W are incident on Li 7 . Then the number of such
gammarays emitted in that direction per unit solid angle and per unit
energy for unit proton flux is just <r py (E, 6, <p) for each lithium nucleus
in the beam. The quantity a P1 is called a differential cross section (per
unit energy and unit solid angle), since it is defined for an infinitesimal
range of the continuous variables E, <p, and 8. The total number of
gammarays emitted in all directions is of course obtained by summing
over all energies and directions. Then the number of gammaquanta
for unit flux and a single target atom is
S P7 = / dE I dip I d(cos 8)<r py (E, <p, 8)
Jo Jo J +1
Here 2 PT would be called the total cross section for proton capture by
lithium for protons of energy W. In this way the frequency of any
process or partial process can be described fully by giving the appropriate
cross section, which may depend on any set of discrete and continuous
1 Note that for the collisions of identical particles, as in the scattering of protons
on protons, incident and recoil particles are not physically distinguishable, and the
number of observed events may then be a sum of both.
Sec. 2B] The Data of Nuclear Reactions 17
variables. The cross section for any process is just the area of the inci
dent beam from which particles are removed by the given process, when
the beam is thought of as directed at a single target nucleus. Of course,
the cross section may be denned not only per nucleus, as we have here
done, but also per unit volume, per gram, or in terms of any convenient
measure of target atoms. If there were no nonclassical wave effects,
the cross section for a definite target, summed over all the possible
processes which could occur, would be just the geometrical area of the
target. We shall see almost this result for the removal of fast neutrons
from a beam (B14, L2, M3, W5).
The probability of a nuclear reaction, expressed either as a cross
section or only roughly as a relative yield, is most frequently measured
for a specific nuclear reaction as a function of the energy of the incident
beam. Such a number is called the excitation function of the reaction
concerned. (If a change in chemical identity is involved, it is sometimes
called a transmutation function.) The chief data of nuclear experiments
besides the energy relations and the nature of the reaction are the angular
distributions and the excitation functions, which can be reduced to a
knowledge of the cross section for the specified reaction as a function of
energy of incidence and direction of emission.
B. The Measurement of Yields and Cross Sections
The experimental procedures for the measurement of cross section
amount to a good fraction of the subject matter of experimental nuclear
physics. It would be presumptuous to try to discuss them in a small
space. It seems worth while, however, to give a kind of summarized
functional enumeration, not so much of the experimental methods, but
of the general procedures and precautions which must be included in any
program for the measurement of cross sections. Evidently the type of
measurements made will depend on the detail and the accuracy of cross
section knowledge wanted; it is obviously more exacting in general to
seek a knowledge of the differential cross section than of the total cross
section, and of an accurate absolute value than of a rough relative one.
1. The Beam. If the cross section a,, t(E, 6,<p) is sought as afunction
of energy, of angle, of type of projectile, it is obvious that the beam
must be well defined with respect to these quantities. Usually this
definition is imposed on the entire incoming beam of projectiles; some
times it is possible to study the reaction products and by some condition
of the reaction eliminate all events which did not begin with the correct
value of one of the variables. These methods frequently involve time
coincidence counting. Generally the entire beam accepted into the re
gion of reaction is known to have the wanted values of its parameters.
18 A Survey of Nuclear Reactions [Pt. VI
The type of projectile is the most general variable which needs
control. The emerging beam from a cyclotron is likely to be quite free
of all types of ions other than the resonant species, but an electrostatic
generator, for example, will equally well accelerate the deuterons and
the protons which may leave its hydrogenion source. Molecular ions
contribute to the current like atomic ions, but not to the yield, and
must often be specifically excluded. For such purposes magnetic resolu
tion of the raw particle beam is often used. When unstable particles
form the beam, as in the case of pimesons, any stretch of beam travel
will allow decay particles to enter the beam. Such beams cannot be
wholly pure, and corrections for the mixture of particles are needed.
Usually charged particle beams are rather easily controlled for beam
purity because of the power of ionoptical analysis. Gammaray beams,
or beams of charged particles energetic enough to yield neutronemitting
reactions in any intervening windows, air, etc., in the path, may require
special care. Crude magnetic analysis will generally remove electrons;
neutrons are much harder to eliminate and may cause trouble. Deflec
tion of the wanted charged beam will in principle solve the neutron
problem, if scattering from the magnets, etc., is not too severe. The
problem is one of increasing importance as beam energy, and with it
the number of secondary reactions, increases.
The energy is often very well defined by the source used. Control of
dc ripple in electrostatic generators has reached precisions of a few
parts in 10 s . Magnetic analysis can be pushed nearly as far even for
roughly defined beams like the cyclotron output. Synchrotrons give
excellent control over beam energy as well. Any matter which the beam
traverses in its path to the target produces energy losses by ionization
and excitation of atoms, and also by nuclear events, which smear out
the beam energy. This is in principle avoidable by keeping the beam in
vacuum, but eventually the beam must strike some target material.
The same processes, of course, occur in the target, which must be kept
thin (measured in atoms per square centimeter) if the cross section is
wanted at a welldefined energy. Thick targets can yield only cross
sections integrated over energy, and may cause straggling in direction
and even particle type as well. Yield as a Junction of energy sometimes
can roughly replace beam homogeneity by allowing a differentiation of
the integral yield.
Homogeneity in energy usually is easier to secure than absolute
knowledge of the energy of the beam employed. The energy scale in
the region of Mev depends on comparison with a few nuclear resonance
energies which have been calibrated absolutely by measuring the trajec
Sec. 2B] The Data of Nuclear Reactions 19
tory of a proton beam in a known electric field. 1 This gives an accuracy
of about one part in a thousand. Magnetic trajectories using the proton
magnetic moment resonance frequency for field measurements work
about as well, and extend into the 100Mev region, especially for elec
trons from gammaray beams. Range measurements using the theory
for energy loss by ionization are very valuable secondary standards over
all energies above, say, a few kev. They have been computed and
calibrated over this whole range for nearly all the usual projectiles.
The angle of emission of the Cerenkov electromagnetic shock wave in
a material of known index of refraction is an elegant method for particles
with a velocity such that j3n > 1, where n is the refractive index of the
material; velocity measurements by direct timeof flight determination
have also been made. The kinematics of elastic collisions is also capable
of giving beam energies expressed as a ratio to the rest energy of the
particle, in the relativistic region.
Angle definition for the beam, often called collimation, is of impor
tance for any measurements of dcr/dQ. Here defining slits of some sort
are the usual key parts of the apparatus. Sometimes the direction of
the track of the product particles themselves, made visible by cloud
chamber or nuclear emulsion techniques, replaces slits followed by
directioninsensitive counters. The effects of Coulomb scattering from
nuclei and electrons of the target itself or any material in the beam path
are the main enemy of good collimation. Divergence of the original
beam forms in itself no source of error which collimating slits or simple
distance of beam travel and choice of detector acceptance angle cannot
remove. But good collimation implies loss of total beam intensity, and
practical limits of time and background here often are decisive.
Perhaps the most important of all parameters of the beam is the
number of particles striking the target in a specified experiment. Often
the current is measured and then timeintegrated. Monitoring, or con
trol of relative intensities, is frequently used for a series of experiments
in which absolute values are either ignored (e.g., all cross sections may
be measured relative to a given known cross section) or calibrated
against one single measurement. Integrated charge may be measured
1 Typical reference energies in absolute volts are
F 19 (p,a T )0 16
0.3404 Mev
Resonance
Li 7 ( P)T )Be 8
0.442
Resonance
F 19 (p,q:t)0 16
0.8735
Resonance
AF(p, T )Si 28
0.9933
Resonance
Li 7 (p,n)Be 7
1.882
Threshold
See, for example, W. Hornyak, T. Lauritsen, P. Morrison, and W. A. Fowler, Revs.
Modern Phys., 22, 291 (1950).
20 A Survey of Nuclear Reactions [Pt. VI
by measuring the potential change or integrated current flow to a Fara
day cup, an insulated conductor so arranged that no secondary charges
can leave it and all beam charge is stopped within it. In one highenergy
proton experiment the Faraday cup was a fortypound chunk of brass
under magnetic field in a vacuum. More usually it is a small cylinder
with a beam window in one end, Ionization is often used to measure
intensity by taking advantage of the known space rate of energy loss
and the mean energy used per ion pair. Direct counting of ionizing
tracks in photoplates can be used. Relative values can be obtained by
actually measuring the beaminduced yield of some known reaction,
using radioactivity, particle emission, or even collecting the neutral
atoms of a stable reaction product, as has been done for such reactions
as IA 6 (n,H s )a. Even calorimetry of a target which stops a totally
absorbed beam has been used for beam intensity.
With particles of nonzero spin (protons or even gammas) there is
another possible parameter to control: the; spin component. This cor
responds to use of a polarized light beam in optical experiments. Apart
from magnetic polarization of neutrons, only the slight polarization
effects resulting from double scattering have been examined for charged
beams. This may become a more important field in the future.
2. The Target. To specify a cross section precisely, the nuclear
composition of the target must be well known. This implies pure and
homogeneous targets. The isotopic concentration of the target must,
of course, be known ; this is usually easy if natural targets are used, but
special analysis is required with targets prepared from isotopeseparated
material.
The target nucleus may undergo the lookedfor reaction, but in gen
eral this is improbable. Most of the target nuclei are not changed by
the beam. Their presence is needed because the probability of reaction
is proportional to the number of nuclei independently exposed. But
their very presence modifies the beam by producing straggling in energy
and direction by Coulomb scattering from electron and nucleus, or by
occasional nuclear collisions of kinds other than that under study.
They may produce a "straggle" in beam particle type by yielding
secondaries which go on in the target to induce unwanted reactions of
nuclei later in the beam path. They clearly modify the beam intensity.
Obviously the use of targets sufficiently thin that the changes in any of
these properties is small compared to the resolution desired in each
parameter is always a solution. But it may be impossible to employ
such a target because the total number of reactions then becomes too
small, or because the physical form of the target elements prevents
preparation of a thin sample. Then a careful study of the beam be
Sec. 2B] The Data of Nuclear Reactions 21
havior in the target material must be made, and corrections made for
all the straggling effects. A frequently employed scheme of finding
crosssection variation with beam energy is to use a target made of a
stack of thin foils, the yield in each of which is measured. The energy
variation of the beam with target thickness tells the mean energy in
each foil, and thus the excitation function is measured. The scheme
demands that the range of the product particles in the target material
be less than the range intervals used in choosing the foil thickness;
otherwise the yield function is partly displaced in a distance and energy
scale, and detail is lost.
Experimental problems of great importance sometimes arise from the
heat dissipation and other physical effects of strong ionizing beams.
Constancy of the target mass and composition is assumed in nearly all
measurements; evaporation, diffusion, coatings from soot of decom
posed pump oils, mechanical displacements of target material by melt
ing or otherwise, and radioactive recoil must all be avoided or corrected
for.
3. The Measurement of Cross Sections. The most direct method of
measurement is an initial absolute measurement of some differential
cross section which defines experimentally the angular acceptance,
particle type, energy, etc., of the reaction product. Then a count of
the number of such events from a fixed target thickness (with correct
tions pointed out above) and with known beam population yields the
cross section directly. The means employed are diverse.
(a) Measuring the Residual Nucleus by Detecting Radioactivity. This
may be connected with a radiochemical separation of that nucleus from
the target matrix. Such a scheme does not define the energy, angle,
spin, or any other property of the product nucleus except its chemical,
or indirectly its nuclear, identity. It may be combined with the thin
layer target scheme to get excitation functions for a specified reaction.
(b) Collecting the Product Particle. If this is done, as usual, while the
product particle is in flight, the angle, energy, and type of product may
all be defined. Very often ionoptical means are used. Such schemes
separate product particles by their differing trajectories in a vacuum
chamber with a controlled electromagnetic field and make possible
detection of individual levels of the residual nucleus by careful meas
urement of the energy released. The chargemass ratios usually fix the
product uniquely. Detectors are used to count the product particles,
which may be linearly responding ionization chambers or solids giving
scintillation response, gasamplifying counters like the proportional and
Geiger counters, photoemulsions, cloud chambers, simple charge collec
tion by Faraday cup, or specific induced nuclear reactions themselves
22 A Survey of Nuclear Reactions [Pt. VI
(W10). The angular resolution is evidently mainly a problem of the
disposition of the apparatus. Energy resolution may be secured by
such ionoptical means as we have mentioned, or by using absorbers to
fix the total energy needed before particle counting. Counter telescopes
which require energy loss rates to be within a fixed range, determined by
signal levels and the coincident or anticoincident response of a fixed
set of counters and absorbers, are very much used at the highenergy
end, where ion optics is more difficult. Cerenkov counters and timeof
flight measurements are among the newest techniques coming into use
in this field.
It is possible to measure total cross section, a = I dil d<r/d&, more
easily. Here what counts is removal from the beam. All that is needed
then is a comparison of detector response with and without the inter
vening target. For thin targets having N atoms/cm 2 and "good geom
etry," then, I/I = e~ N ". "Good geometry" means such collimation
that all particles scattered through any small angle miss the detectors
(B14). Since the detectors have finite acceptance in every case, correc
tion is needed for scatteringin (P3). When this correction is sizable,
the case is said to be one of bad geometry. Thick targets, where multi
ple collisions are not negligible, require additional care. The inescapable
relation in Eq. (45) between an absorption cross section and elastic
scattering means that special care must be taken to correct for the very
lowangle forward coherent scattering if the geometry is very good,
since scatteringin cannot be estimated from the angular distribution
of the main absorptive process. Apart from such questions, it is in
general possible to make the only absolute crosssection measurement
one of total cross section, done by beam attenuation, and then only
relative measurements for all the differential cross sections are needed.
4. Special Problems with GammaRays. The use of gammarays as
beam projectile, or their measurement as product, has special problems
arising from the small interaction with matter. Gammarays can be
detected only after they have given their energy at least in part to mat
ter, especially to electrons. For this purpose most gammaraydetecting
schemes make use of a layer of matter of appropriate thickness placed
ahead of the detector proper. In this converter layer the gammarays
produce electrons (or positrons) whose ionization, range, energy, and
momentum may be measured by the means described above. At ener
gies in the range of a few Mev it is possible to get absolute measure
ments of gammaray flux by using the known ionization of the electrons
secondary to gammas, which of course have a spread of energies as long
as Compton scattering is important. At energies above 10 Mev or so,
Sec. 2B] The Data of Nuclear Reactions 23
a thin foil in which electronpositron pairs are the principal products
may be used. Then a knowledge of the pair production cross section
from theory or from total attenuation measurements may be used to
compute the incident flux. Calorimetry is possible for strong beams
of not too high energy, and special photonuclear processes may also
play some part, particularly as secondary standards. Radioactive sources
of gammarays, or gammaray sources which use a nuclear radiative
transition like the decay of the familiar state of Be 8 at 17.6 Mev excita
tion, may be calibrated by using other particles emitted in coincidence,
if the details of the disintegration scheme are known. The strong sources
are usually gammarays produced by the stopping of electron beams by
Coulomb scattering, or bremsstrahlung radiation. This radiation is of
course continuous, rather than discrete, like that of the nuclear radiative
decays. Its energy limit is in general wellknown, because the original
electron energy can be welldefined and measured. Individual quanta
may be selected by using coincidence methods with the electron after the
radiative act. In general, though, the whole spectrum is used, or at least
that above some threshold for a given reaction, or above a broad limit
set by absorbers placed in the beam to remove the softest quanta. In
this case cross sections are hard to give for a definite energy. Again dif
ferential measurements may be made. Very often the cross sections for
bremsstrahleninduced reactions are given not per quantum of a definite
energy, but per effective quantum in the beam. The number of effective
quanta Q is denned (B16a) as the ratio Q = II n{E)E dE /2? maX)
where n(E) dE is the number of quanta in the beam exposure having
energy between E and E + dE and E max is the maximum energy in the
beam. The rough form of the spectrum for bremsstrahlen is n(E) dE
= N v dE/E; therefore in this approximation the value of Q for a brems
strahlung beam is just the number N . This is usually satisfactory if
the region of interest is well above the lowenergy deviations from the
l/E distribution for the beam. For quantitative work the exact form
of the bremsstrahlung spectrum must be taken into account and the
experimental data analyzed by a numerical process indicated, for ex
ample, in (K4).
Neutrons present many related problems; their detailed treatment is
to be found in Part VII.
5. A Table of Yields. As orientation to the size of yields to be ex
pected in typical practical situations, Table 1 shows a number of thick
target yield values for various nuclear projectiles of widely available
energies. In all cases the target is taken to be thick enough to stop the
entire beam, and to be the pure element of normal isotope composition.
24
A Survey of Nuclear Reactions
[Pt. VI
For gammarays and neutrons such a target is not usually practical,
but computed values are given for comparison. Only a few values are
given for reactions due to other particles than the heavy charged parti
cles of central interest in this chapter. Table 7 contains additional data
Yields of
TABLE 1
Some Typical Nuclear Reactions (ThickTarget)
Product Nuclei/10 6 Projec
Projectile
Reaction
tiles (beam energy in Mev)
Gammarays
AF(Y,P)
8 X 10 s (17 Mev)
AF(t,w)
3 X 10 3 (17 Mev)
Cu(7,n)
11 X 10 3 (17 Mev)
W( Tl n)
18 X 10 3 (17 Mev)
Electrons
Cu 65 (e,e'n)
2 (17 Mev)
Neutrons
ncapture on any but He 4
10 6
S 32 (n,p)
~2 X 10 5 (5.8 Mev)
Protons
Cu(p,n)Zn 63
18 (6.3 Mev)
Deuterons
Be(d,H 3 )Be 8
230 H 3 (14 Mev)
Li(d,2n)Be 7
22 (19 Mev)
B(d,n)C u
1.5 (8 Mev)
C(d,n)N 13
1.4 (8 Mev)
Na(d,p)Na 24
190 (19 Mev)
P(d,p)P 32
88 (8 Mev) ; 350 (14 Mev)
K(d,p)K 42
11 (19 Mev)
Ca(d,a)K 42
0.3 (19 Mev)
Cu(d,p)Cu 64
330 (14 Mev)
Y(d,2n)Zr 89
4.9 (14 Mev); 52 (19 Mev)
Mo(d,a)Cb 90
0.4 (14 Mev)
Te(d,n)I 131
1.6(8 Mev); 32 (14 Mev)
Bi(d,n)Po 210
56 (14 Mev)
Alphaparticles
A(a,pn)K 42
17 (44 Mev)
Pb(a,2n)Po 210
48 (44 Mev)
Bi(a,2n)At 211
0.13 (38 Mev); 130 (44 Mev)
on deuteron reactions. Reference (GO) contains still more data on
charged particle beams and a very useful survey of chemical methods
for preparation of carrierfree radio isotopes.
C. Types of Reactions : A Guide to the Literature
It is not the function of this section to present a complete survey of the
thousandodd nuclear reactions which have been studied. Even a list
of reactions by types cannot be exhaustive, since reactions have been
observed at high energy in which dozens of products occur. It would
be possible to systematize such highly multiple reactions only by giving
the relative yields of the various products. Putting aside such processes
Sec. 3A] The Nuclear Model 25
for the present, most nuclear reactions involve the emission of two
particles after the reaction, following the type of reaction T(i,p)Ji. A
sizable number of reactions are known in which three or four definite
products are emitted; then there is a continuous sequence of more
complex reactions which typically occur at high energy. There is also
the familiar fission reaction, in which, instead of a rather heavy nucleus
emitting several light nuclear particles, the heavy target divides and
two heavy fragments come off, accompanied by several neutrons and,
infrequently, an alphaparticle. A systematic list of nuclear projectiles,
with the products they induce, and references to original literature
which describes reactions of the type so defined are presented in Ap
pendix II. The references make no attempt at completeness, nor have
they been chosen for the purpose of giving the first papers on the reac
tions. They are meant to indicate a few recent and rather complete
papers, especially those which contain theoretical treatment and good
bibliographies. It is hoped that entries in Appendix II will assist the
reader both in finding a rather general introduction to the reaction type
of his interest and in getting a good start toward an exhaustive search
of the literature.
SECTION 3. THE NUCLEAR MODEL
A. Qualitative Account of the Model
We can now begin to construct a nuclear model which will have some
connection with the complex reality of nuclear process and structure.
In the study of nuclear reactions the comparison of the detailed implica
tions of successive models with experiment has led to a graphic but still
by no means adequate model. It is this model we here seek to describe.
It is instructive to compare our knowledge of the atom with that of the
nucleus at its heart. We know at last that the atom is fully described
as an assembly of rather welldefined electrons, held together by their
electrostatic attraction to the dense central nucleus, influencing each
other by their mutual electromagnetic interaction, and moving in accord
with the principles of quantum mechanics (especially the exclusion
principle). The nature and properties of electrons are wellknown, all
interactions being given in detail by Maxwell's and Lorentz' equations.
Only for very few details — the hyperfine structure and isotope shift,
the line shift due to electromagnetic radiation field coupling — is further
investigation needed. Moreover, the wellmarked periodicities and shell
structure are themselves evidence that the complex mechanics of a
manybody problem are susceptible in the atomic case to farreaching
26 A Survey of Nuclear Reactions [Pt. VI
simplifications in which the manybody problem is soluble by methods
which begin with the idea of the independent motion of a single electron.
Contrast the nucleus. Even the forces which act between pairs of
nucleons are still without any general expression. In Part IV of Volume
I, the twobody problem is discussed at length in an effort to find empiri
cal clues to the force laws. The nuclear forces are shortrange forces;
no largescale counterpart exists for laboratory investigation. All
our knowledge of nuclear forces is based on nuclear experiment itself.
The nucleus is a system of many bodies, whose general mechanics is
evidently more complicated than that of the equivalent atom for,
although regularities exist in nuclei, they are not so striking as those
evident in the periodic table of the atoms. The major regularities,
important though they are (see Part IV, Volume I), cannot yet lead to
confidence in a simple buildingup process like that of the atomic domain.
Only in the lightest nuclei, below neon, or even below helium, is more or
less easy progress to be hoped for in obtaining a definite and detailed
picture of nuclear matter. We shall see how the shortrange and
exchange character of the forces, the strong component of noncentral
force, the whole tightly bound character of the nuclear state of matter
make difficult its detailed description. The point of view so far most
successful in the discussion of reactions, though clearly itself incomplete
and often misleading, involves a kind of abstraction from the detailed
structure of any particular nuclear species, and an effort to see what
types of behavior, what experimentally determined properties, can be
used to give an account of nuclear reactions.
1. The Interaction of Nucleons. In the atom, each electron moves
in a rather slowly changing force field, the sum of the nuclear attraction
and the averaged repulsion of the other atomic electrons. The forces
are of longrange nature, and it does not matter much if we neglect the
very occasional close collisions between the other electrons and the one
whose motion we are considering. We can hope to study the motion of
each electron separately, taking account of the others present only in a
general and smoothedover way. The presence even of scores of electrons
does not impossibly complicate the problem : we fix on the detailed mo
tion of each in turn, correcting only slightly for the specific behavior of
its neighbors. This approximation, known as the Hartree approxima
tion, is excellent for the loose, smooth, open structure of the atom, gov
erned by longrange forces. But in nuclei the scheme seems entirely
inappropriate. Here the forces are of a range smaller than the dimen
sions of the system. The force on each particle depends strongly on the
detailed configuration of its neighbors; a small difference in the position
of another nucleon can change its interaction from a decisive one to ;i,
Sec. 3A] The Nuclear Model 27
negligible one. The miniature solar system has disappeared; we think
instead of a miniature drop of water, a small drop of quantummechanical
liquid. Instead of a slowly and smoothly varying mutual interaction,
superimposed as a perturbation upon a stronger central force, all of
which we can without much error replace by a smoothedover time
average, the interaction energy of a single nuclear particle with the rest
of the nucleus is a jagged and highly fluctuating function of time. To
average in space over such a function is to conceal its essential features.
The continual transfer of energy from particle to particle is the rule;
no wellmarked center of force exists. It is misleading to try to "peel
off the motion of each particle" in turn and try to correct for the average
effect of its neighbors.
In this picture the shell structure of loosely coupled particles has no
obvious counterpart in nuclei. Yet the lightest nuclei show such proper
ties that for a long while they have been studied by the use of the
Hartree approximation and related methods. Moreover, the nuclei
all the way up the table exhibit regularities, in their lowest states of
excitation at least, which imply a welldefined shell structure. Such
properties as (1) special stability of particular values of A and Z (of
which the familiar increased stability of eveneven nuclei is the proto
type), (2) the angular momentum and parity of the ground state and,
for many nuclei, of the lowestlying excited states (isomeric states), and
(3) rough values of the magnetic moments of ground states can be
rationalized and even predicted by semiempirical results of what is
called the shell model.
In the most recent and remarkably successful of the shell models (M5),
the individual nucleons are pictured as moving in HartreeIike shells of
welldefined orbital angular momentum and radial quantum number.
The order and the total angular momentum of these states can be fixed
by arguments based on the picture of strong spinorbit forces, which
couple the spin and orbit of each individual nucleon strongly together,
so that the total angular momentum and parity can be established by
the analogue of spectroscopic jj coupling, taken with certain semi
empirical rules for deciding the order of the terms. There results the
order of shells, in the familiar spectroscopic notation, shown in Table 2.
There is a rapidly growing literature of applications of the shell model
to the prediction of the properties of lowlying nuclear levels; it cannot
be doubted that no previous effort to describe such states has been so
successful, and indeed it seems demonstrated that strong spinorbit
forces must be present.
How the singleparticle orbit model, with its marked shells, can be
reconciled with our general arguments on the unsuitability of the
28 A Survey of Nuclear Reactions [Pt. VI
TABLE 2
Nucleab Shell Oeder in the jj Coupling Model
The suffix indicating the j value has been suppressed, except when the same
orbital angular momentum appears in two adjoining shells; then the state of
highest j is indicated as lying lower in energy. The symbol 2p, for example,
means both the states 2p% and 2p^.
Shell
Configurations
i
Values
Number of
Nucleons
in Shell
Total Number
(Neutron and
proton shells
are independent)
Is
lp
Id, 2s
1/, 2^, lff9 /4
lgt/i, 2d, 3s, \hm
Xh%, 2/, 3p, liijs
A
A, A
A, A, A
A, A, A, A, A
A, A, A, A, !K
A, A, A, A, A, X A
2
6
12
30
32
44
2
8
20
50
82
126
Hartree picture for the strong, shortrange, and fluctuating nuclear
forces is still by no means clear. But an interesting analogy due to
Weisskopf seems to point out the nature of the eventual solution. He
recalls the behavior of the Fermi gas of electrons, regarded as noninter
acting, and moving in the periodic potential of a metal lattice. Here
shells — the Brillouin zones — are very marked, and the neglect of the
electron Coulomb interaction, certainly reasonably strong, seems to
have very little effect. Qualitatively this must be ascribed to the circum
stance that electronelectron collisions can make no physical difference
if all states in the momentum space are filled, as required by the exclusion
principle for the Fermi gas. Such scatterings represent no change in the
total system. If, however, a higherenergy electron enters the metal
from without, it carries momentum adequate to excite the electrons
of the Fermi gas beyond the Fermi limit; its collisions are rapid, and it
loses energy very quickly to the electrons of the metal. In the same
way, the lowlying states of the nuclear system may be represented well
enough by the noninteracting particle picture; the expected collisions
are mostly excluded. Yet for high excitations the regions of momentum
space which are not filled can be reached, and the collisions become
decisive. This picture fits the real nuclear situation very well.
Although the stationary states of nuclei, and especially of the lighter
nuclei, lie beyond the scope of this treatment, it is appropriate to men
Sec. 3A] The Nuclear Model 29
tion here the most powerful of the methods deriving from the Hartree
picture, which has found great use in the discussion of just those prob
lems. The method is the Wigner theory of nuclear supermultiplets.
Using an extension of the same essentially grouptheoretic methods
which permit the classification of the spin and orbital quantum states
of the atomic system, Wigner considers a nuclear system a collection of
particles interacting by forces which depend only on spatial configura
tion, not on spins or on the charge of the nucleon. Exchange forces may
occur, if they are spaceexchange forces only. For such interactions,
certain quantum numbers can be defined, in extension of the J and S
numbers, familiar in atomic multiplets, which characterize the spatial
symmetry of the nuclear wave functions for any collection of nucleons.
Then, by regarding the Coulomb and the spindependent forces as
perturbations capable of splitting the highly degenerate levels of the
ideal system, a more realistic picture can be obtained which has con
siderable success in the ordering of the lowerlying states of not too
complex nuclei. For specific information the extensive literature may
be consulted.
However, it is fundamental to the picture of the nucleus which we
shall chiefly employ, the picture appropriate for excited states: that
energy exchange with ease between the closely packed nucleons; that
the nucleus be regarded as a highly condensed state of matter, a tightly
packed quantum liquid, where the relatively small number of degrees
of freedom (compared to those of a drop of water), and the diffrac
tion and exchange effects to be expected from quantum considerations
at such small distances be thought of as modifying the behavior of a
system otherwise very like the thermodynamic system of a familiar
liquid drop. We know that the radius of the sphere of nuclear matter
is rather well represented by the formula
R = r A y ° (23)
with A the number of nucleons, and r = 1.4 X 10~ 13 cm (CIO). This
is just the assertion of constant density which defines our model as a
liquid or a solid, not a gas.
From the simplest Fermi gas model, or what is almost the same, by
taking the mean velocity of a nucleon in the nucleus to be that corre
sponding to a de Broglie wavelength equal to the mean spacing of
nucleons, we find that the kinetic energy of a nucleon ought to be some
tens of millions of volts. This gives a mean collision time of some 10 — 13
cm per 5 • 10 9 cm/sec, or about 10 22 sec. For any interacting particle
which takes a time longer than this to complete its collision with a bound
nucleon, we have to regard the nucleus as a closed system, without a
30 A Survey of Nuclear Reactions [Pt. VI
sharp separation possible between collisions with a single nucleon and
collisions with the whole of the nuclear matter. Only if collisions are
made by particles moving so fast, and transferring so much momentum
to the struck nucleon, that the collision is complete within the charac
teristic nuclear time can we think of the nucleus as a collection of free
particles. Even then the nuclear "gas" is highly degenerate; quantum
levels are filled up to a rather high energy, and low momentum transfers
thus are discouraged. It is then clear that, for incident particles with
an energy per nuclear mass unit up to 20 or 30 Mev, the nuclear drop
must be considered as a whole. Only in the higherenergy domain are
we more nearly justified in thinking of a collection of instantaneously
free nucleons; and even here there are important effects of the nuclear
binding.
Let us continue this qualitative discussion of the collision between a
nucleon and the struck nucleus. If the incident nucleon is of high enough
energy so that it may be localized (i.e., represented by a wave packet
whose main components have wavelengths smaller than nuclear dimen
sions), we may try to follow the collision classically. The shortrange
nuclear forces extend about as far as the mean spacing of nucleons within
the structure. A colliding particle then has very little chance to travel
through the nucleus without striking the nucleons within. At very high
energies (say 100 Mev or more) the mean free path for collision with
the nucleons of the nucleus has dropped to some 4 X 10~ 13 cm, but,
using the same rough idea of collisions with nearly free neutrons, the
free path has dropped to a tenth of that value for tenmillionvolt
incident particles, and to only 10~ 14 cm at 1 Mev (S7). The last dis
tance is so small that the picture evidently fails, but the conclusion seems
confirmed. As soon as the particle crosses the surface of the nuclear
sphere, it will interact strongly with the nuclear matter. It may lose
only part of its energy to the first particles it encounters, and continue
on, transferring energy to the nuclear matter as it goes. Eventually its
entire kinetic energy is spread in some way over the many particles of
the nucleus, and a new state of quantized collective motion for all the
nuclear particles exists, a nearly stationary state of what is called the
compound nucleus. We shall say more of this in Section 3A2.
The nuclear drop has of course an energy content, given in an under
standable form by the semiempirical formula discussed in Section 3B4.
The electromagnetic properties of the nuclear drop may be computed
and measured; indeed, all the properties of the ground state can be
ascribed to the model. But the details of nuclear structure cannot be
obtained from such simple assumptions as underlie the semiempirical
energy formula, especially the constant density of nuclear matter.
Sec. 3A] The Nuclear Model 31
Still less is to be expected from the use of this statistical model in nuclear
reactions. In general, it may be said that what we here visualize is not
the particular quantum level of a particular nucleus with a welldefined
energy, but the average to be expected in a given region of mass number,
including many similar but not identical neighboring nuclei. More
formally, we shall deal with all the characteristic properties of nuclear
states not one by one, but averaged over an interval of excitation energy.
Only if the fluctuations in these properties are not too great from level
to level, so that the average behavior represents the individual behavior
of states, or if the experimental situation produces averages as the quan
tities directly observed, can we hope for satisfaction from such a statisti
cal point of view. It is in this charitable sense that the results of the
theory are to be taken.
We can summarize the above statements by setting out the two leading
assumptions under which the results of the present theory of nuclear
reactions are obtained:
1. Nuclear matter is composed of closely packed particles, strongly
interacting and capable of rapid interchange of energy, like the particles
of a drop of some quantummechanical liquid. We may call this
assumption close packing.
II. The results obtained from such a consideration are meant to
apply not to the particular and specific properties of one welldefined
state of motion, but to an average over many states of neighboring
energies. Only if the property studied does not fluctuate too widely
in such an interval will the conclusions be trustworthy. This we may
call the statistical assumption.
2. The Compound Nucleus. In Section 3A1 we discussed the dissi
pation of the kinetic energy of some incoming particle as it passed into
and merged with the closely packed particles of the nucleus. The
resulting configuration we called the "compound nucleus." Let us go
on with the story of such an event. After the incident particle has
merged completely with the nucleus, its energy, no longer concentrated
in one particle, is shared by the collective motion of all particles of the
new system: the compound nucleus formed by projectile and target.
Each of the nucleons will have some additional energy, but none will be
likely to have all the incident energy at any one time, or even any very
large fraction of it. The energy will be spread among the very many
degrees of freedom exactly as the heat energy of a drop of water is kinetic
energy spread among many degrees of freedom. But the compound
nucleus will eventually lose its energy of excitation. Either radiation,
which, as we shall see, is a relatively slow process, will finally remove the
energy and thus "cool" the nucleus, or after a long time (time enough
32 A Survey of Nuclear Reactions [Pt. VI
for many collisions of nucleons within the nucleus, some ten million in
typical cases) the configuration will have the rather unusual property
that a sizable amount of energy is concentrated on a single particle, and
a particle may escape, cooling the nucleus by "evaporation." It is
evident that the escaping particle will not in general be the particle that
entered, and that it will have neither the energy nor the direction of the
incident particle. Even if by chance a neutron, say, goes out when a
neutron enters, this need not be an elastic collision. On the contrary,
excitation energy will in general be left behind, to be dissipated in another
way; the collision will be inelastic. Elastic collisions do not take place
with formation of a compound nucleus except by the reemission de
scribed under the special and rare circumstance that the residual nucleus
is left in the single ground state only. The elastic scattering observed
includes other effects by which the incident wave function is scattered
coherently without the actual formation of a compound nucleus, through
purely wavemechanical processes.
The formation of a compound nucleus may, it is true, take place
under rather special circumstances. A gammaquantum, with wave
length long compared to the nuclear size, may excite proton vibrations
with a definite phase relationship. Or, considering the same process
from a quite different point of view, the quantum may act upon a single
surface proton which receives enough energy to escape without striking
other nucleons on its way out. Or the proton coming out may lose
energy to a single spot on the nuclear surface, "heating" the nucleus
not as a whole but locally. The final reaction might occur by evaporation
before the "conduction" of heat away from the hot spot to the rest of
the nucleus has taken place. In all these cases — we believe them infre
quent but possible — a detailed consideration of the particular process
(B17, Kl) would be required. However, we shall discuss mainly the
fundamental notion of Bohr, which is that the compound nucleus is
capable of disintegration in a manner independent of its method of
formation. This is perhaps the third assumption of our theory, and we
have seen under what circumstances it would require modification. We
shall state it explicitly:
III. Nuclear reactions in the domain of statistical theory take place
in two separable steps: (1) the formation of a compound nucleus by
combination of the incident particle and the target nucleus, and (2) the
subsequent disintegration of the compound nucleus, in a manner inde
pendent of its method of formation, into the products of the reaction,
the emitted particle and the residual nucleus in some one of its quantized
energy states. A reaction that leads to more than two products proceeds
by a continuation of this scheme: the first product comes off, and the
Sec. 3A] The Nuclear Model 33
residual nucleus then acts as an excited compound nucleus, emitting
another particle, and subsequently even another, and so on. Thus
as many as a dozen neutrons may be "boiled off" successively from a
highly excited nucleus (T4).
3. Nuclear Energy Levels and Level Widths. So far we have dis
cussed the ideas of the statistical theory from the point of view of the
closepacked nucleons, an almost geometrical and quite classical argu
ment. The same conclusions can be expressed in terms of energy levels
rather than in geometric terms. We have already shown that the com
pound nucleus exists for a time long compared to the mean free time for
a nucleon collision in the nucleus. This means that, although the descrip
tion of the individual nucleon as free with a definite energy was very
rough, the compound state can be thought of as existing with a wave
function which is the approximate eigenfunction of the Hamiltonian
of all the nucleons. We restrict ourselves to states which genuinely
represent the compound nucleus or its disintegration products: that is,
to wave functions which contain no parts corresponding to one particle
at a distance from all the others, except where the distant particle is
represented by an outgoing spherical wave. This serves to exclude
states in which a free particle exists which could not have been emitted
by the compound state. Under these conditions we can write
*. = ^ n e [E "^ ir " /2)m (24)
where the energy is no longer a real eigenvalue but contains an imaginary
part. This is the familiar procedure for a damped oscillator (H6).
The fraction of systems in the state n will vary with time like the factor
e IW \ Thus T n /h is the reciprocal mean lifetime of the state; V n is
called its width. The usual expression of the uncertainty principle
AE • At c^ If is exemplified here.
What does the spectrum of a nucleus look like? Simply by considering
the distribution of energy among the many particles of a tightly bound
system we can form some idea. We begin with the fact that the binding
energy of a particle in the nucleus is only a few times smaller than the
rough estimate we can make of its kinetic energy. If the particles did
not interact strongly — a bad assumption — we could think of forming
the excited levels of the nucleus by giving all the excitation energy to
one nucleon, or dividing it among two, three, • • • , and so on. Now,
since the well in which we could imagine the particles to be moving (see
Fig. 3) is far from fully occupied with particles, we can find levels for
each nucleon which lie quite close to its ground level. Thus the dis
tribution of a few Mev of excitation energy could be accomplished in a
very great number of different ways. Most of these would correspond
34
A Survey of Nuclear Reactions
[Pt. VI
to the general spreading out of the excitation energy among the many
nucleons; very few indeed would be the states in which all the energy
was used to promote one or even a few nucleons. The coupling of the
particles of course means that the levels are of mixed character. A level
cannot be ascribed to a specific distribution of energy, spread among
a definite set of particles. In any state, part of the energy is sometimes
concentrated on one, sometimes on two, etc. Each eigenfunction could
be expanded into a series of terms corresponding to the various partitions
of energy. But, with so many possibilities available, the contribution
E
Continuum
(dissociation)
rz — ^
:l=5S?s=:
_ ~ ~
Empty levels
> Filled levels
■
y
Effective
potential
well
Fig. 3. Schematic nuclear potential well and nucleon energy levels.
of a partition which corresponds to assigning all the energy to one or
even two nucleons will be very small indeed. The total number of levels
will evidently be extremely large.
These levels, as we have said, are not completely sharp. Only the
ground level is quite sharp (even it is widened imperceptibly by the
possibility of beta or alphadecay or of spontaneous fission). As we
add excitation energy, the levels become wider because of possible
radiative transitions to the lower states through emission of quanta.
For some sufficiently high excitation energy, the nucleus will be capable
of emitting one particular particle, say a neutron. This will, of course,
represent a widening of the levels. But, since so little of the wave func
tion represents a configuration with all the available energy concentrated
in one neutron, the effect will be by no means large. The level width
will not change markedly as a new decay process appears. The levels
will simply widen progressively as more and more possibilities of decay
become energetically allowable. The character of the levels will not
change abruptly with their energy. Only the appearance of a small
probability of emission of a particular particle will indicate the crossing
of the energy threshold. As the energy of excitation increases, the total
Sec. 3A]
The Nuclear Model
35
width of levels rises. Moreover, the increase with energy of the number
of ways the energy can be partitioned is marked. Finally, the spacing
is so small and the width so large that the levels overlap and the familiar
continuum has been reached. Even here fluctuations in level density
and special selection rules may maintain the features of a discrete
spectrum. Only when the excitation is comparable to the total binding
energy — say 6 to 8 Mev per particle — does the character of the levels
change qualitatively. In such a region the concentration of sufficient
t
> Continuum
(dissociation)
A few empty levels
Atomic
potential
well
Fig. 4. Schematic atomic potential well and electron energy levels.
energy in one nucleon to allow its escape is the rule and not the exception.
Under these circumstances a genuine continuum of levels exists.
In Fig. 4 is shown the parallel situation in the case of an atom with
many particles. From the geometrical point of view, the atom is a loose
open structure. Its longrange forces mean that elastic collisions are
the rule; even if an incident particle excites the atom, the energy of
excitation is given to one or perhaps a few of the valence electrons. The
K shell electrons have in a heavy atom about a hundred thousand volts
of binding; the valence electrons, only a few. Even the L electrons are
bound only by some tens of kilovolts. The possible energy levels thus
are as shown in the figure: very narrow bands of close levels with great
empty spaces separating the bands. Thus in general there is little
chance that the energy of an incoming electron lies in a region where
levels of the compound system exist. For atomic collisions these nearly
stationary states are of little importance. Almost any system which
has energy enough to release an electron will rapidly do so. It is inter
esting, however, that, even in a system so simple as the beryllium atom,
terms have been found which give rise to broadened lines. These terms
correspond to states in which the total excitation of the atom is greater
36 A Survey of Nuclear Reactions [Pt. VI
than the ionization potential but is divided between two electrons.
These states often have large widths and short lifetimes, because they
can decay by radiationless transitions, emitting electrons. The analogy
to the nuclear case is close. In molecules, of course, predissociation is a
prominent effect.
Although the concepts of level spacing and level width are entirely
quantummechanical, Weisskopf has suggested a very pretty semi
classical picture of their significance. Consider a complicated many
particle system like a nucleus, but for simplicity with a large number of
equally spaced levels, with energy of the nth level, E n = E + nS. We
can make up a wave packet by combining a great number of energy
eigenf unctions <p„(r) corresponding to the stationary states E n . The
total wave function is then
N / N \
; V „ iEn*/t I V" 1 „ inSt/h\
t = 2, anVne =1 L a n <p n e J
iBof/k
Now it is evident that
/ 2xh\ 2 .
( 2xh'
Thus we have constructed a wave function which is by no means a sta
tionary state; in it the configuration of the system repeats itself after
the lapse of a time t, which we can call the recurrence time. If the level
spacing is 5, the recurrence time is given by the relation t = 2irh./8. Of
course, a real nucleus has levels which are not equally spaced, but a sim
ilar physical interpretation may be given to the quantity D, the mean
level spacing at a given excitation energy. Closely spaced levels imply
long times of recurrence, and conversely. It is also instructive to con
sider the distance traveled by an identifiable nucleon of the wave packet
during the duration of one period of the motion. If the mean velocity
of the nucleon is given by something like a Fermi gas model, chiefly
the zeropoint motion arising from the confinement within the nuclear
volume, we get for the velocity v = h/MpF. Then the path length I
traveled by a nucleon which had, say, just penetrated the nuclear surface
from without before it found itself back in the original configuration
again would be I = 2irh 2 /MDp F . Taking a level spacing of 10 ev,
appropriate for the region of slowneutron capture in heavy nuclei, the
value of I ~ 10 — 6 cm, which is about a million times the nuclear diam
eter. This makes picturesque the complex motion in a compound state,
and adds some confidence to our ideas about the statistical theory.
On the other hand, if D = 1 Mev for the lowlying states of light nuclei,
Sec. 3A] The Nuclear Model 37
the recurrence path is only a dozen diameters or so, and the picture of
an orbital motion for a single particle becomes plausible once more.
The level width can be given a similar interpretation. If a state has
energy enough to emit a nucleon, we can imagine the packet so built
as to describe the configuration existing when a given nucleon has just
entered the nuclear surface. Now, after the recurrence time t, the
nucleon would be back at the surface again, with the same energy, ready
to leave in just the manner in which it entered. If it were actually to
leave at this first recurrence, the implication would be that the lifetime
of the state, h/r, was just equal to the recurrence time. Then the spac
ing and width would be of the same order, and the idea of welldefined
compound state hard to justify. This is plainly one limiting case. But
we must recall that at the nuclear surface there is a sharp change of
potential. Within the surface, the strong nuclear forces act; outside
(a distance only a couple of times 10 13 cm outside), there are no nuclear
forces at all. At this potential jump, the particle de Broglie wave suffers
reflection. Suppose that the probability for penetration of the barrier
is P. Then the width can be given in terms of the recurrence time:
T ~ h/(r/P); and hence, in terms of the level spacing, T ~ PD/2t.
For the case in which the barrier is entirely nuclear (no Coulomb or
centrifugal forces, realized by an s wave neutron), the penetrability is
simply given by the familiar quantum formula P = 4k/K, where the
quantities K and k are, respectively, the wave numbers for the nucleon
de Broglie wave within and without the nuclear surface, and we have
gone to the limit k « K. This is a very important result of our detailed
theory, to be discussed later at length. Of course, in the present account
we have disregarded the fact that there are alternative modes of decay,
and the fact that there are constants of the motion besides energy, like
angular momentum and parity, which must be conserved but which may
vary from level to level. In the calculation of the appropriate D, such
classes of states are to be regarded as contributing independently. In
general, the statistical theory we discuss here is limited to the treatment
of levels en bloc, without much hope of identifying states of a specific
kind or of predicting energy levels. Since it is clear that the many well
defined observed levels in thermal neutron absorption in the heavy ele
ments (spaced as they empirically are some 10 ev, and lying 8 Mev above
ground) must be not far from millionth members of some series of terms,
it is not unexpected that we would make more progress, in heavy nuclei
at least, by some general statistical treatment than by a method based
on classifying and identifying individual terms.
Individual levels can be observed at low energies in all nuclei, e.g.,
wherever gammaray spectra can be studied. In the lighter nuclei, such
38
A Survey of Nuclear Reactions
[Pt. VI
levels are visible up to tens of millions of volts of excitation; in heavier
nuclei, not more than a million volts beyond the neutron binding energy.
The most general features of observed level distribution is the steady
increase of level density with excitation energy by a function which
becomes the steeper as we go to heavier nuclei. A few examples of
10'
_ / V N 1511.7
hJ3
a)
ft
>
CD
10 ;
Observed averages for
zero excitation
A
F 206.7
Na 247.2
~ Al 287.9
8Mev
Observed points:
ZA excitation energy
Curve models:
—  ■■ ■ Semiempirical
 — — «— Fermi gas
_  — Combinatorial
o
100
200
Fig. 5. Nuclear level spacing. This figure presents the nuclear level spacing as a
function of the mass number A for various nuclear excitation energies. The marked
curves are plots of the results of various models discussed in the text. The circles
mark individual observed levels; the nucleus and its excitation are marked beside
each experimental point. The three horizontal lines show the spacing observed at
zero excitation, averaged over a fair number of nuclei in each region of A. The
roughly satisfactory nature of the theoretical curves is shown, as well as their inade
quacy for both light nuclei and heavy nuclei, like Ni and Pb in which the influence
of shell structure is marked, especially at modest excitation energy.
estimated average spacing, taken from the few known levels in the
various regions of interest, are given in Fig. 5. It is noteworthy that the
average spacing near zero excitation decreases from the lightest to the
heaviest nuclei by a small factor, perhaps by about a factor of 10, while
the spacing at some 8 Mev excitation changes by five orders of magni
tude. It will, of course, not escape the reader that the application of the
statistical notion of level density to the few levels of the lightest nuclei
is a dubious business.
Sec. 3B] The Nuclear Model 39
The width of the levels increases as their spacing decreases. We
expect levels that overlap one another in particle excitation of reasonably
heavy nuclei some 10 Mev above the ground state. Unfortunately the
difficulty of producing closely monoenergetic beams with high kinetic
energy has limited the possibilities of experimental check. The use of
endothermic neutronyielding reactions has been about the only direct
test of this point. The work of Barschall with total neutron cross sec
tions at energies around \ Mev on many elements indeed serves notice
that this picture is too simple (B3). The levels vary greatly in spacing
and strength ; for certain nuclei the whole level structure is far too coarse
grained for the validity of our picture. This is primarily a result of the
fact that the shell structure strongly modifies the binding energy of the
added neutron. Excitation by neutron capture with low kinetic energy
may excite some nuclei (those with closed shells) by only a fraction of the
usual 8 Mev. At the lower excitations, the level density is much reduced.
Other reasons may also be involved.
B. Calculation of Level Densities
Again disregarding certain details of structure, we may proceed to
calculate mean level densities for some statistical models of the nucleus.
We shall present two of the most instructive in some detail, giving only
a sketchy account of some other schemes which have been proposed.
1. Combinatorial Model. We have already discussed qualitatively
the simple model for which we here calculate the mean level density.
It is based upon the idea of a set of independent particles among which
we divide the energy of excitation. Suppose that each of the N particles
has an identical energy spectrum, which we shall take to be of the sim
plest possible form: a set of equidistant levels of uniform spacing 8.
Then the excitation energy W can be written as
W = £ m&
i
where the n t are integers showing which level of its own spectrum each
particle occupies. A given energy W can then be realized in as many
different ways as there are ways to sum integers to a given integer total.
The states thus formed would be highly degenerate, the degree of de
generacy being equal to the number of ways in which the fixed sum can
be realized. If we now take into account the interaction of the particles,
the degeneracy will be split, and each level will be split into as many
levels of the whole system as the value of its degree of degeneracy. In
this way the level density can be calculated. The result is based upon
the asymptotic formula of Hardy and Ramanujan (H4) for the number
40 A Survey of Nuclear Reactions [Pt. VI
of different ways, p(n), to form a given number n by any of the possible
sums of smaller integers:
p( n )=_J_ „*<*»>*
4(3) ^n
If we take as a value for the unit level spacing 5 the average observed
separation between the lowest levels of heavy nuclei, 2 • 10 5 ev, we get,
for the degree of degeneracy expected for a total excitation energy W
of 8 Mev,
Q
pi—) = p(40) = 410 4
which gives an average level separation for the whole excited nucleus
of about 5 ev, which very roughly corresponds to the density of level
separation estimated from slowneutron collisions. The level separation,
4.(3)^We~' iH ' w/s)>i
p(W/sy
depends exponentially on the excitation energy; this is characteristic
of all nuclear models and is displayed even by this very simple one.
2. GasLike Models. In this model the nucleus is visualized, in the
first approximation, as consisting of gas — a set of noninteracting par
ticles — held in a spherical box of nuclear size. The two Fermi gases
present (neutrons and protons) occupy all the lowest levels in the box.
As the nucleus is excited — "heated" — some of the higher levels are
occupied with increasing excitation. Bethe has given a careful statistical
calculation for this model, following the usual methods of statistical
mechanics (B13). He has used a statistical ensemble with energy,
particle number, and angular momentum taken as given constants.
The calculation is of course quite parallel to the calculation of the entropy
of the system, since entropy is proportional to the logarithm of the
number of states of the system. Somewhat greater care is required for
the computation of the number of states than for its logarithm. If W
is the energy of excitation and A the number of nucleons, he obtains,
for the mean spacing D (in ev) of nuclear levels of some definite angular
momentum /,
D = 5 .ltf . (^}\ eUw/2.1)" (25)
In this formula we have assumed the usual r = 1.5 • 10"
is in Mev.
Sec. 3B] The Nuclear Model 41
A similar but more sophisticated model was proposed by Bardeen (B2).
He solved essentially the same problem, but he made use of the proper
ties of the exchange forces in nuclei. It has been shown that the two
body problem in a square well with exchange forces could be reduced
to the onebody problem except that the potential acting on one particle
now depends on the wave number of the interacting particle. It turns
out that the total energy of the equivalent particle motion increases
more strongly with the wave number than the kinetic energy alone.
This effectively increases the phase volume available for levels, reduces
the number of individual particle states, and therefore much more
strongly reduces the density of levels of the whole system. The effect
is quite analogous to the effect of the correlations of electrons which
obey the exclusion principle in changing the interaction energy of elec
trons in a metallic lattice. The model of Bardeen is often referred to as
the freeparticle model with correlations. The energy dependence of
the result is given by Eq. (25) with (ATT/2. 1) replaced by (ATT/4).
3. The Liquid Drop. The idea of regarding the nucleus as a classical
drop, with volume energy and surface tension, is by now a familiar one.
On this basis the normal modes of vibration of a liquid drop may be
computed: there are surface vibrations and transverse and longitudinal
volume waves. The surface energy and the compressibility of the
nuclear matter may be estimated from the semiempirical energy content
given in Section 10. Then the vibrations of each normal mode are
thought of as quantized, and energy distributed among them according
to Planck's distribution law. The nuclear matter is so "cool" that
equipartition is far from achieved; the excitation energy is not very large
compared to the zeropoint energy. From the treatment of Bethe,
numerical results can be obtained (B13). Here again the energy de
pendence is given roughly by the formula
Z)~exp (cTT M ), TT small D ~ exp (c'TT M ), TT large (26)
It is interesting that in this treatment one can show that only a few
normal modes of surface vibration are present for heavy nuclei in the
region of 1Mev excitation or lower. The possibility of rotation of the
drop, associated of course with its angular momentum and the multipole
moments of its lowlying states, has been considered. Though the idea
is a satisfying one, it can be said that the drop model does not account
quantitatively for the lowlying levels of high angular momentum which
seem to be responsible for nuclear isomerism. On the contrary, direct
calculation of the spacing of rotational levels gives for their energy, for
a lead nucleus say, only a few kilovolts. Such fine structure has not yet
42 A Survey of Nuclear Reactions [Pt. VI
been observed. It has been pointed out that particles whose wave func
tions must satisfy symmetry properties may, unless very rigidly bound,
be unable to rotate as a rigid body rotates because some rotational
states of motion are not allowed. Indeed, it has been shown (T3) that
only for high angular momenta, of the order of 50 units, will nuclear
rotational levels have the order normal for a rigid body or a classical
drop rotating as a whole. At such angular momenta, the excitation
energy is some Mev, and vibrational motion is of course important.
The suggestion that a drop or a solid be used to represent the lowest
states of nuclei has not proved very satisfactory; as a matter of fact, the
magnetic moments of ground states are much better represented by the
idea of an odd particle rotating about a nonrotating core containing
all but the odd nucleon (S4). Again, we may expect better results for
the drop model in highly excited states than near ground. A formulation
of the structure of nuclei, where individual level properties are decisive,
is still beyond the scope of the present models, although for the lowest
states the individualparticle shell model in jj coupling form has had
remarkable success.
4. Thermodynamic and SemiEmpirical Results. The general behav
ior of the level density as a function of energy is similar in all the models
discussed. The details are not correct, and the constants deduced com
pare indifferently with experiment. It seems reasonable then to compute
a general form of the level density as a function of energy from the most
general statistical or "thermodynamical" point of view, and to adjust
the constants until we get reasonable agreement with the not very well
known level spacings observed.
Let us begin with the fundamental statisticalmechanical relation for
the sum over states (B13)
z=Z.
Ei/r
where the values E t are the energies of the states of the whole system,
and r is the parameter which we can identify with the usual kT, the
temperature measured in units of energy. Since we are dealing with
dense levels, we can rewrite the sum as an integral :
Z=Jp(E)e~ }s ^dE (27)
But let us introduce the function A, the Helmholtz free energy of thermo
dynamics, by the relation: Z = e~ AlT . Now, if the system has a large
number of degrees of freedom and the distribution is one of equilibrium,
we expect that the contributions to the integral in (27) come mainly
Sec. 3B] The Nuclear Model 43
from the region near the maximum, which we call W, the mean excita
tion energy of the nuclear system. We then rewrite the expression as
Z = Ce ln '> ElT dE
/a(lnp) a'lnp (igWO 2 i (gWQ
e al (E_,,0+ 8^ 2 + ; rf#
Expanding the integrand about its maximum by the familiar saddle
point method, we obtain
w , T /a 2 (ln P £?/r) 1*
Z = P (W)e^/' • 2x/ * \ = e (28)
L / ah \wi
Now let us introduce the familiar thermodynamic function, the entropy
S, by the relation
A = W  tS (29a)
We expect this function to be proportional to the logarithm of the den
sity of states. From (29a) we get the elementary definition of entropy :
dS 1
=  (29b)
dW r
and we can use this in (28) with d(lnp)/dE \ W = 1/t, to get
e S(W)
o(W) =
T{2wdW/drY A
We then have, for the general connection between density of states
p = 1/D and the thermal properties of the system,
J."
dW'/r
P (W) = v (30)
K t(2tt dW/dr) 1A
To determine the level density for any system, then, we need to find the
entropy of the system, which we can always do by integrating (29b),
if we know only the expression for the heat content as a function of
temperature, that is, the relation between excitation energy and the
temperature parameter t(W).
Next, the procedure is either to adopt a model, and so work out the
heat content as we have indicated in the previous sections, or to make
some general assumptions for the temperature dependence of the heat
content, and then fit the constants to the data.
We can proceed in a general way by using thermodynamic arguments.
We expect the heat capacity for nearly any system to be a steadily
44 A Survey of Nuclear Reactions [Pt. VI
increasing function of the temperature. If the temperature of a system
is so high that for all the degrees of freedom the energy r is large com
pared to the quantum energy ho> , the classical limit of equipartition is
reached, and the heat capacity is constant. But the nucleus, as we have
said before, is generally rather "cold." The simple model of the Fermi
gas of A nucleons in volume f ttR 3 yields the result (M4) that the "gas"
is degenerate if the excitation energy is small compared to Ah 2 /2MR 2
~ 8A H Mev. Then heavy nuclei are highly degenerate below excitation
energies of some 50 Mev. The interaction of nucleons does not affect
this conclusion, just as the electron gas in metals is degenerate in spite
of the electronelectron forces.
Now we can write the heat content as a function of temperature in
the power series:
W = c + c lT + c 2 t 2 j
From the general quantum ideas of the third law of thermodynamics,
we expect the specific heat to vanish as T — > 0. Thus the expansion of
W near zero temperature, i.e., in the strongly degenerate region, ought
to begin with at least a quadratic term, with all higher powers negligible.
The familiar Debye specific heat for metals x actually vanishes at low
temperatures with an even higher power of the temperature, since the
heat content goes to zero like 2 14 . If we assume then any simple power
law dependence, W = cr n , we can integrate
dS _dS dr 1
dW~ drdW~ t
letting
n
S =
(n1)
With this expression, we obtain for p(W) the value
> m " ^ (r^)"' 2 «"■' (<dhj ' c "" w """)
For the Fermi gas, the exponent contains 1 — K, for the liquid drop
the power W* to W H at low and high temperatures respectively.
It seems appropriate to use the very simple form
P = Cexp(aJF)^ (31 a )
If we do this, and try to make a rough fit to the data, we obtain (following
Weisskopf) (W5) for nuclei heavier than A = 60 a result which is approx
1 See (M4), p. 253, for formula; p. 248 for derivation.
Sec. 4] Nuclear Level Widths 45
imated by the simple form, for nuclei of odd or even Z and N — A — Z,
1 12
~ ^oddodd ~ ("evenodd ~ ^ oddeven = *b eveneven = T~. T (JVIev)
2 (.440)
a = 3.4(A  40)^(Mev) 1 (31b)
We present in Fig. 5 a few selected data on observed level spacing,
together with the values calculated from the three nuclear models
described. There is some inconsistency in what the figures represent,
for the experimental data from slowneutron capture, as well as the
freeparticle model calculation, refer to levels of a definite angular mo
mentum (or within ±Jh from the ground state). The thermodynamic
calculation certainly includes all levels. Here there is a chance to add
some more definite assertion to the general statistical notions; all we
can do at present is regard all levels as more or less uniformly distributed
in regard to most not very large values of angular momenta. This
vague statement is typical of our statistical theory, and it may account
for discrepancies observed in individual cases where specific angular
momentum properties may be involved. The least we can do to recog
nize the great importance of nuclear energy shells is to use differing
constants for odd or evenZ nuclei.
SECTION 4. NUCLEAR LEVEL WIDTHS
We have already mentioned the concept of level width, which is defined
as r„ in the expression for the approximately timeindependent (because
longlived) quasistationary state wave function for the compound
nucleus,
From this form for the decaying amplitude of the wave function the
width can be related to the mean lifetime in the usual way: r n = h/i„.
This expression naturally refers to the change in amplitude with time
of the compound state regardless of the products of its decay; it is the
total width, and r n is the total lifetime. Naturally, the compound state
may in general decay in more than one way; a single set of reactants
may have many alternative sets of products. We may generalize the
notion of width to include the idea of partial width by writing T n
= 2 rVa(w), where the r ta are the partial widths, which we index here,
ia
somewhat redundantly, with two indices: i, which designates the kind
of particle emitted, as neutron, gamma, proton, etc.; and the index a,
which gives the state of the residual nucleus. It is evident that the same
46 A Survey of Nuclear Reactions [Pt. VI
particle can be emitted with different properties, as different energy or
angular momentum, leaving behind generally, though not necessarily,
a different state of the residual nucleus. The argument of the function
Ti a (ri) refers to the state n of the compound nucleus whose width is here
written. Clearly the partial width for the emission of, say, a neutron
to any possible state of the residual nucleus might be the experimentally
useful quantity, and it would involve the sum ^ r neutpa (n) over all
a.neut
accessible states which can be left behind after a neutron is emitted.
To speak somewhat academically, r„ is strictly zero for almost no
states of nuclei. Even the ground state is not wholly stable; in radio
active nuclei it may decay by beta or alphaemission, and in nuclei not
observed to be radioactive we know that in general a state of less energy
content can be reached either by a process like spontaneous fission or,
if the nucleus is a rather light one, by combination with a neighboring
nucleus of the matter present. Such lifetimes are long, perhaps enor
mously long, compared to actual nuclear reaction times, under terrestrial
conditions at least. It is conventional therefore to disregard such forms
of radioactive decay, and especially betadecay, and to regard the levels
as widened only by the possibility of radiation or by the emission of a
heavy particle. This implies that, while I^ may be different from zero,
its study belongs properly to radioactivity theory, and not to the dynam
ics of nuclear reactions. The levels of a typical nucleus then can decay
only by gammaemission, which slowly widens the states as their excita
tion increases until the excitation energy reaches a value which makes
heavyparticle emission energetically possible. For nuclei whose ground
states are empirically alpharadioactive, of course, this condition is ful
filled for every level, but for most nuclei from the middle to the top of the
periodic table such a value is reached only at the binding energy of a
proton or neutron. The binding energy lies near 8 Mev for the entire
middle of the table; for the lightest nuclei, up to neon, say, it is widely
varying; for the nuclei from gold or lead and up, it has fallen to 5 or
6 Mev (W3). Up to a level of the compound nucleus, then, which has
an energy content at least 5 to 8 Mev above the ground state, only
gammaradiation will introduce a finite width; beyond it, first the
emission of one particle, then of another, and finally of even more kinds
and a greater number of particles will widen each level. Moreover, not
only do the values of T(n) increase because there are more possible I\,
but in addition more states are available in the residual nucleus, so that a
takes on more possible values. The widths themselves increase slowly,
as we shall see, with the kinetic energy of the emitted particles and the
energy of the gammarays.
Sec. 4A] Nuclear Level Widths 47
A. Level Widths and Reaction Cross Sections : Statistical Relations
We have already indicated that the fundamental idea underlying the
present statistical treatment of nuclear reactions is the separation of the
process into two independent steps: the formation of the compound
nucleus, and its subsequent independent decay. Upon the basis of this
assumption there can be derived a useful relation between level widths
and cross sections. The method of the derivation is the familiar statis
ticalmechanical one of detailed balancing (W5).
Let us consider an equilibrium mixture of three kinds of systems: the
compound nucleus we are studying, and its two decay products, the
"emitted particle" and the residual nucleus. The equilibrium reactions
are the decay of the compound, and its formation by recombination of
the two decay products. We may write
C* *± p + R a
We do not consider every possible state of the residual nucleus R and
the particle p which might combine to form the given state of the com
pound nucleus, but only particles of such energy, angular momentum,
etc., that they can combine with a residual nucleus ~R a in a definite state
a to produce a particular state of the compound nucleus C*. It is clear
that we must be sure of a random statistical ensemble: we must be cer
tain that special phase relations do not exist between the wave functions
of the decay products and the compound nucleus which emits them or is
formed from their combination. To insure this, and to free ourselves
from the possible special character of individual states of the compound
nucleus, we have to regard the levels of the entire system as very closely
spaced, and be sure that the energy interval is large enough to include
many levels of the system. Our widths and cross sections are then aver
ages over this small interval of energy, small compared to the scale of
variation we are interested in, but large enough to include many levels.
All phase relations are then lost in the averaging. This is .the usual
procedure required to define a quantumstatistical ensemble.
Now we apply the almost revered formula of the timedependent
perturbation theory, which gives the transition probability per unit
time for the transition from the almost stationary state (A) of a system
to a continuum of states B at the same energy. This is
2tt . .,
w A ,B •= —\Mab\pb®
h
where [ Mab \ 2 is the square of the matrix element of the perturbing
operator between the initial and final states involved (the final state is
48 A Survey of Nuclear Reactions [Pt. VI
chosen as any one of the continuum having the correct energy), and pb
is the density of states of the final system, per unit energy in the volume
within which the wave functions are normalized. Now the inverse
process is given by the same formula written for w B ^ a But, since
the operator is certainly Hermitean,  Mab \ 2 = \ M B a  2  We then
obtain the expression for detailed balancing in its general form
WA ^BPA = W B ~,A'PB (32)
The use of perturbation theory conceals the generality of the result; any
order of the perturbation will give the same dependence on the absolute
square of a Hermitian matrix element, so that the result is quite general.
Applying it to the situation described above, with a compound nucleus
of excitation energy W decaying to two reaction products, we may write
Tpa(W) 4TP 2 dp
— p c (W) = Vffpaga. — — • — 
n (2irhy dW
Here we have used the relation between width and lifetime, w = 1/V
= r/h, and found p B by using the density of states for a free par
ticle with mass m p , momentum p, velocity v, and kinetic energy E p ,
47rp 2 /(2irh) 3 , and converting the density in momentum space to the
density in energy by the factor dp/dW. Note also the important relation
w = v<x for any collision between two particles of relative velocity v,
in normalizing volume. The factor gr R is the statistical weight (degree
of degeneracy) of the residual nuclear state. We may finally write
T pa (W ) PC (W) =^~ with X = — (33)
2jt 7rX m p v
This formula tells us that the mean level width, measured in units of the
level spacing, is just equal to the cross section for the inverse reaction,
a pa , measured in units of the maximum cross section for absorption of
particles of the given kinetic energy, apart from small statistical weight
factors. It is interesting to recall that the maximum absorption cross
section for s waves is just 4irX 2 , and thus the width for s wave emission
is at most 2g R /w times the level spacing.
The use of the detailed balancing formula for connecting lifetimes to
cross sections as demonstrated here is only one of its applications. By
considering an overall reaction i + T — » p + R, two cross sections
may be similarly connected. In general,
viTPi 2 = <r P RP P 2
where p; and p p are respectively the initial and final relative momenta.
Sec. 4B] Nuclear Level Widths 49
In (33) we could have defined T pa and p c somewhat freely. They
might contain averages over all levels in a narrow band of energy, or
they might be restricted to levels of a single class, e.g., only those for
which the angular momentum involved could in fact be given to the
particle p. Since the level densities are not accurately known, such
distinctions are ordinarily not carefully made.
B. Calculation of Level Widths
The assumption of the independent decay of the compound nucleus
can be exploited in detail for the calculation of level widths. With this
idea we combine the evident fact that the forces on nuclear particles
which operate at large distances from the nucleus — distances large com
pared to the range of nuclear forces — are, we believe, completely known.
These forces are, of course, (1) the very important Coulomb force on
all charged particles, and (2) the effect of angular momentum, the
centrifugal force associated with nonvanishing angular momentum.
Most of the success of the statistical theory really comes from the fact
that nuclear reactions are greatly affected by these essentially non
nuclear forces, and separation and calculation of their consequences is
itself a considerable step toward explanation of the most evident phe
nomena of the reactions.
We shall show that under suitable conditions the partial width for
emission of a definite particle with given orbital angular momentum
I and energy E P) leaving behind a particular state of the residual nucleus,
can be written as the product of three factors:
T p ia(W n ) = ky(n)P pl (Ep) (34a)
where k is mv/h. for particle p, y(n) is a factor which depends only upon
the state of the compound nucleus, particularly its excitation energy
W n , and P P i is the usual coefficient of penetrability of whatever barrier
the particle encounters from electrostatic or centrifugal forces.
Let us begin by writing the total width of a given state of the com
pound nucleus, which we shall index by T(n), as a sum over the partial
widths, one for emission of each kind of particle indexed by i, with angu
lar momentum Z, leaving the residual nucleus in state a. [Compare
(33) and (34a, b).] This implies that the wave equation H^ n = W n ^ n
for the whole system has been solved by the wave function V n = 2p a fo,
where by i we index all possible emitted particles, for which ^ }  is the wave
function outside the nucleus, and <p a is the wave function of the corre
sponding residual nucleus in state a. We obtain a set of discrete but
widened levels by imposing this condition on the solution, with the
specification that the function \pi contain only outgoing and no incoming
50 A Survey of Nuclear Reactions [Pt. VI
spherical waves. Below the energy for which some particle can be
emitted, the function ^; will correspond to a bound state for that particle;
above the threshold, to an outgoing spherical wave. It may be noted
that, just as the presence of nonzero T(n) means that the states of the
compound nucleus are broadened, and that their amplitude decays, so
the complex W„ = E n — iT n /2 results in a complex wave number k p
for the outgoing particle. The amplitude of the outgoing particle wave
function is greater at large distances, since the particles represented as
farthest from the nucleus were emitted earliest, when the compound
nucleus had its largest amplitude. The asymptotic expression for the
radial part of an outgoing wave e ikr /r will be damped by the presence of
an imaginary term in k. Below the threshold for emission, this is the
only term. Above the threshold, we can neglect this imaginary part so
long as the width T n is small compared to the kinetic energy of the
emitted particle. With this in mind, we write
Yi m (6,<p)R pl (r p )
*p = 2,
l,m r v
where R p i/r p is the radial part of the wave function of particle p, and
Y e m is the normalized spherical harmonic.
1. Barrier Penetration. The value of a partial width T p i{n) is just
h/rj,;, and hence just h times the number of particles emitted per second
from one compound nucleus. We normalize R p i by the condition that
I  'Apa \ 2 dV — 1, where the integral is extended over the spatial
i/nucl
region "within the nucleus," say out to a fixed distance R. This implies
that T P i is small, so that we can neglect the more remote part of the
wave function of the compound state. This is closely related to our
general assumptions. Then the width is just h times the outgoing flux
of particles integrated over a distant sphere of radius r x . We obtain
r h 2
T p i(W„) = hrj v x \ f pl {r x )  2 <Kl = — • k\ R P i  2 (34b)
J m p
We have required that very far away, beyond a radius r x , the radial
dependence of R p i be simply e z ^ kr+s) . This means that we have chosen
r x so large that in that region the kinetic energy h. 2 k 2 /2m is very large
compared to the electrostatic or centrifugal barriers. From this point
on, the flux of outgoing particles falls off only geometrically, with 1/r 2 .
Hence the width becomes independent of r. But, in the region between
the radius R which marks the boundary of the compound nucleus, and
r w , which is the forcefree region, the behavior of the wave function is
Sec. 4B] Nuclear Level Widths 51
fully known. It is attenuated by the requirement of penetration through
a potential barrier, which may be zero for neutrons of zero orbital angu
lar momentum, but would be very considerable for alphaparticles of
any orbital angular momentum if the kinetic energy were not high com
pared to the Coulomb energy, and would be appreciable even for neutral
particles of nonzero angular momentum. We shall write R p i(r x ) in
terms of the value of R P i{R) at the nuclear surface R. The quantity
P p i, which we call the penetrability of the barrier, we define as the
fraction of the initial intensity of particles, of the type indexed by p and
angular momentum I, with kinetic energy E, which penetrates to a
fieldfree region. We can write the Schrodinger equation in the field
containing region for the radial variation only:
\ d 2 2m P/ 1(1+1)1
If the solution F pt is that which corresponds to a unit outgoing wave,
with asymptotic value e +i(  kr+i) , then evidently
_ external intensity 1
intensity at nuclear surface  F v i(R) \ 2
For uncharged particles the solutions F pi are well known to be the
halfintegerorder Bessel functions. The boundary conditions are satis
fied by the socalled Hankel function of the first kind,
M?r
H l+ y 2 w (kr)
A few penetrabilities for uncharged particles of various angular momenta
can be given:
Pnfl = 1 Pn.l = — — "T P,
x 2 ~ 4
n,2
1 + x 2 "■' x * + 3x 2 + 9
x 6
n ' 3 ~ ?+te* + 4& a + 225 (36a)
with x = kR. And, for general I values, if the wavelength is long or I
large, i.e., if x — kR <JC I
= x»
n ' 1 ~ (21  l)\2l ~ 3) 2 (2l  5) 2 • • • l 2 (36b)
A table of such functions has been published (Ul).
The calculation of Coulomb barriers is much more difficult, because
of the wellknown slow convergence of the asymptotic expressions for
52 A Survey of Nuclear Reactions [Pt. VI
Coulomb wave functions. There are two approaches : an exact calcula
tion can be made by using the published tables of Coulomb wave func
tions (Y5). Such a calculation has been made and graphs published
for exact penetrabilities for protons for various angular momenta emitted
from the lightest nuclei, from Li to F (C5). The other method is the
familiar approximate solution for the wave equation in the classical
limit. This is the W.K.B. method, useful under conditions in which
the distance traversed by the particle while its wavelength changes appre
ciably is many wavelengths. If the usual graph of the energy barrier is
recalled, it will be seen that the approximation is good only when the
energy of the particle does not come near the top of the barrier. This
means that the W.K.B. method is useful either in the trivial case far
above the barrier when the penetrability is evidently unity, or in the
very useful case of low penetrability. In the transition region, for P of
the order of unity, the W.K.B. approximation is not reliable. The
relevant formulas will be given here, though they are common in the
literature.
The Coulomb penetrability from the W.K.B. approximation can be
written
Ppi(E) = ^—  1) exp (2C,) (37a)
Here Bi is the barrier height for a particle of charge ze, reduced mass
m — nipM/nip + M, and angular momentum I; Ze is the nuclear charge
and R the nuclear radius. The quantity Ci is the familiar phase integral
/2m\ H r r >
Ci = [ — ) J (VE)^dr (37b)
where the potential energy, including Coulomb and centrifugal terms, is
zZe 2 1(1 + l)h 2
V(r) = + ~ and V(r e )  E = 0; B t = V(R) for any I
r 2mr
The value for Ci is complicated:
Ci 1 /tt .12*
1 Ar 1  2» \
— = — I — f arc sin — 1 — (1 + y — x)
C A 2(x)^\2 (1 + 431/)*/
H
/2zZe 2 RmV A 2(x) H \2 (1 + Axy)'
(1+2(2/)%* +(1 + 2/ x)*])
+ (y) log ^Ti^ (37c)
with y = 1(1+ l)/(2zZe 2 Rm/h 2 ), and x = E/B t . A plot of Ci has
been published for a sizable range of x and y; when used with the other
Sec. 4B]
Nuclear Level Widths
53
graphs in Appendix I, it makes the necessarily long computations as
convenient as can be expected.
It is plain from formulas (37) that the orbital angular momentum of
the partial wave concerned plays a large part in determining the ampli
tude at the nuclear surface. High orbital momenta mean higher centrifu
gal barriers, and the penetrability through the centrifugal barrier may
be small for high I, even when the simple Coulomb barrier has been sur
mounted. It is useful to plot a few barrier heights to show where this
approximation may be expected to be satisfactory. We assume the usual
formula R = r A* with r = 1.5 X 10 13 cm.
TABLE 3
Barrier Heights (in Mev) tor Various Particles and Angular Momenta f
Nucleus
Angular
Particle
Mo
Ne 10 20
Ca 2 „ 40
Zn 30 66
Sn 60 112
Yb 70 m
Th 90 232
Emitted
menta
V
3.5
5.6
7.1
10.0
12.1
14.1
1
6.1
7.2
8.3
10.8
12.7
14.5
2
11.1
10.4
10.6
12.4
13.8
15.5
3
18.7
15.2
14.0
14.8
15.6
17.0
d
3.5
5.6
7.1
10.0
12.1
14.1
1
4.8
6.4
7.7
10.4
12.4
14.3
2
7.3
8.0
8.8
11.1
13.0
14.8
3
11.1
10.4
10.6
12.3
13.9
15.5
a
7.1
11.2
14.3
20.0
24.1
28.1
1
7.7
11.6
14.6
20.1
24.3
28.2
2
9.0
12.4
15.2
20.5
24.5
28.5
3
10.9
13.6
16.0
21.1
25.0
28.8
t These barrier heights were computed by using R = 1.5 X 10 13 A 1 ^ cm.
Now we can collect the results of the last paragraphs. Using (34b)
and (35), we can write
h 2
T P i(W) = — * R p i(R) \ 2 P p i{E)
(38)
We identify this expression with our formula (34a), if we set for the
factor G
h 2
m v
R P i(R)
(39)
54 A Survey of Nuclear Reactions [Pt. VI
It is evident that the effect of the external barrier is contained wholly
in Ppi) we can call the product ky p i the "width without barrier." The
argument of the function R p i contains explicitly the nuclear radius, and
the function represents the motion of the emitted particle within the .
reach of the specific nuclear forces. It is clearly a function of the internal
state of the compound nucleus, as we had expected for y. Strictly
speaking, it will also depend on the wave function and its derivatives
outside the nucleus, on the kind of particle, and so on, for the function
R P i must be a continuous solution of the wave equation. It is the point
of the statistical treatment to ignore the dependence of 7 on the external
part of the wave function. Especially if we remember that the nuclear
forces are strong, that the radius R is rather welldefined though not
sharply so, and that we should try to apply the theory to the average
behavior of levels and not to specific ones, will we accept this assertion.
A more formal argument has been given, but it cannot be made air
tight because of the very nature of the theory. It seems more satis
factory to accept this simplification in the spirit of the main Bohr
assumption: that the compound nucleus decays in ways independent
of its mode of formation and of the details of single levels.
SECTION 5. THE COURSE OF NUCLEAR REACTIONS
A. The Steps of the Reaction
We can now proceed to the discussion of the nuclear reaction as a
whole. The process of formation of a compound nucleus and its subse
quent disintegration can be described by a reaction cross section written
this way: „ ._. t _ ,,„.
J <j iv = Si(E)HD p (40)
Here Si is a cross section for reaching the nuclear matter, a process sup
posed to depend only on extranuclear forces; £ is a number of the order
of unity which is called the sticking probability. It is roughly the
probability that the incident particle will enter to form the compound
nucleus once it has reached the nuclear radius. The factor f will carry
our uncertainty about the validity of the statistical assumptions; it is
in this factor that we can throw all the features about the phase relation
ship of the possible decay products which form part of the wave function
of the compound state. If the nucleus is excited only at a "hot spot,"
where the lifetime of the whole compound nucleus is long compared to
the time of possible emission of particles from the locally excited region
of only a few nucleons, we expect the sticking probability to decrease.
In general, the more nearly independent of energy and exact reaction % is,
the more satisfactory is the statistical model.
Sec. 5B] The Course of Nuclear Reactions 55
We have written as a separate factor, again in the spirit of our ruling
assumptions, the probability D p of disintegration of the compound
nucleus, once formed, into the particle p. This can, of course, be
written in terms of the various partial widths:
Dp = tj< (41)
where the denominator contains a term for all the possible products of
the disintegration of the compound state. The possibility of alternative
reactions, and their relationships as the incident particle type and energy
vary, is evidently expressed by the changing magnitudes of the various
T's.
B. The Contact Cross Section
1. Elastic Scattering and Diffraction Effects. The value of <j pl will, of
course, not really be a smooth function of energy. It will reflect, through
a variation in D p , the presence of resonances where they are well defined.
For energies high enough so that the classical idea of the collision can
be applied, but not so high that the collision times for bound nucleons
are long compared to the time of interaction of the incoming particle,
the simplest picture of the compound nucleus would lead us to think of £
as approaching unity: every particle that comes to the surface of the
nucleus — as well as that can be defined — will stick to form the compound
state. We should like to compute the factor Si by referring as completely
as possible to the region outside the nuclear volume. This is of course
not rigorously possible. The incoming wave which represents the inci
dent particle will be modified by the presence of the nucleus in many
ways. Those scattered components which are not coherent with the
incoming wave, but which represent inelastic scattering or the occur
rence of a genuine nuclear reaction, can all be referred to events within
the compound nucleus. There are scattered coherent waves which
represent the actual formation of a compound nucleus, with the subse
quent chance emission of a particle of the original energy and type.
Much more important coherent scattered components, however, come
from the effects of the extranuclear forces, from the diffraction by the
geometrical surface of the nucleus, and from the "reflection" by the
nuclear potential. These cannot be distinguished physically in any way
from the elastic scattering following compound nucleus formation. The
inverse process alone provides a means for evaluating the probability
of such elastic scattering by compound nucleus formation, and thus a
precise definition for <S,.
56 A Survey of Nuclear Reactions [Pt. VI
Let us write the usual expression 1 for the wave function of an incident
plane wave of particles of wave number k = mv/h., where m is the mass
and v the velocity of the incoming beam in the centerofmass rest
system. This expression is a sum of partial waves written in spherical
coordinates, each corresponding to a given value of the orbital angular
momentum. We normalize to an incoming flux of unity by writing
* = ju **" = —^u Z (21 + l)^i l J l+ y 2 {kr)Y l \e) (42a)
(v) /2 (2krv) /2 1 =0
Here Ji+y 2 is the Bessel function, and Y° the normalized spherical
harmonic. At great distances from the scattering center we obtain
f _* (lY . 1 J2 (21 + l)X.il.( e «*r+lW2) _ e N(*r+I,/2) )Fj (42b)
\tv kr o
This is in the absence of any effect of the scatterer. But the scatterer
may change in general the phase and amplitude of the outgoing part
of the wave at large distances. Suppose that the relative phase and
amplitude of the outgoing wave after scattering is given by the complex
number 77. Clearly  17  is at most 1. Then the wave function in the
presence of the scatterer is no longer given by (42b) but by a sum of
incoming and outgoing waves proportional to e~ lkr and e +lkr respec
tively. The incoming portion is
fcn = S*in (,) , *in W  ()" ^ (2* + 1) *6«*'+"' 2 > F,<°> (43a)
\v/ kr
and the outgoing
^out = 2>/w W , *out (I) = ( T X( (21 + l)*(+„)e+«*'+"'»r I < , »
\v/ kr
(43b)
For a particular value of I the number of particles absorbed per second
is just the difference between incoming and outgoing flux evaluated over
a large sphere:
r 2 f[v\ hn w ! 2  v\ +oJ l)  2 ] dSl = ^ (21 + 1)(1   „  2 ) (44a)
Since we normalized the incoming wave to unit flux in the beam, the
cross section for absorption is given by
•Tabs = 2ff ab . (,) <7 ab s W = J (21 + 1)(1  I i,  2 ) (44b)
1 This whole treatment follows closely that of (F7).
Sec. 5B] The Course of Nuclear Reactions 57
The elastically scattered wave has to be added to a plane wave in
order to get the actual perturbed wave ^ in + ^ out . This is simply
lAin + i^out — e* 70) H  Again the flux in this wave integrated over a
large sphere gives the elastic scattering cross section:
a ei = W° ffel « = ^ (21 + 1)(1 + v )\ 2 (44c)
It is valuable to consider the relations between these waves. The
total cross section <r tot = 2<r tot w can be written:
<rtot W = <r e l (0 + <r abs W =—(21+ 1)[1 + jfcfo)] (45)
/c
There is a wellknown and interesting result which follows from this
formula. Consider the cross section of an obstacle in the classical limit,
with the radius (R) of the scatterer very large compared to X = 1/k.
We expect the classical absorption cross section, summed of course
over all values of I to obtain the total cross section, to be wR 2 if the object
is "black" i.e., if it absorbs every particle whose trajectory strikes it.
But in this case v = 0, and the total cross section <r tot = 2wR 2 . Where
does the additional ttR 2 of elastic cross section arise? This is the familiar
"shadow scattering," now observed repeatedly for neutrons of 90 Mev
and less on heavy nuclei (CIO). The black sphere clearly casts a shadow
m the beam. This shadow must, on the wave picture, as in the familiar
Kirchhoff treatment of physical optics, be produced as the result of
interference between the incident wave and a scattered wave. The
total intensity of scattering, to produce a shadow the size of the object,
evidently corresponds to a cross section just equal to the geometrical
one, tR 2 . Thus there must be an elastic cross section of irR 2 . This
can be observed because, especially in the nuclear case, the shadow does
not extend to infinity even in a parallel incident beam. On the contrary,
diffraction by the sphere means that the direction of the scattered wave
is changed by an angle of the order of X/R. This has the consequence
that the shadow is dissolved by the diffracted beam in a distance of
R 2 /X from the scattering center, and the elastic scattering can there
be observed.
The angular distribution of the shadow scattering is easy to compute
for a black sphere, in the limit X/R « 1. The result is that
da(8, <p) _ R 2 J x 2 (kR sin 8)
da ~ ~ shTf? (46) •
where k = 1/X = h/M v , R is the nuclear radius, and <p the scattering
58 A Survey of Nuclear Reactions [Pt. VI
angles, and.Ji(z) the Bessel function of the order of unity. Actually the
contributions of partial waves with I ~ R/X are small but complicated.
If K/R ~ 1, the effect of important values of I will not be given correctly
by (46), and the distribution and magnitude of the elastic shadow scatter
ing will be more complicated. The success of this result has suggested a
nuclear model of optical type in which the nuclear volume is regarded
as a sphere of "gray," not black, material which both absorbs the inci
dent wave in part and disperses it, as a result of the phase shift due to
the mean potential which the incident nucleon feels in passage. This
model has given interesting results in the highenergy region, which is
discussed in more detail in Section 11.
It is worth pointing out that the maximum value of Caba^ is (21+ l)wX 2 ,
and that this maximum can be reached only if y = 0. The maximum
value of <r e \ w is (21 + l)47rX 2 and can be attained only when y = +1.
There is in general a range of values of tr e i (i) possible for each value of
(Tabs^. That the maximum scattering cross section is four times the
maximum capture cross section reflects the fact that maximum scattering
comes from just reversing the phase of the incoming wave to form the
scattered outgoing wave. This has the effect of adding to the plane
wave just twice its outgoing part, while complete absorption simply
removes the outgoing portion of the plane wave. Thus the cross section
for scattering depends on the square of twice the amplitude correspond
ing to complete absorption, which gives four times the cross section.
2. Influence of Extranuclear Forces. We have so far considered only
maximum and minimum values for scattering and absorption, obtained
by assuming values for the quantity r\. It is clear that a real calculation
of 17 would require the solution of the manybody nuclear problem, 1
and this is precisely what we are unable to do. But we can try to bring
explicitly into view the effect of extranuclear forces, reserving £ to de
scribe the effect of the specific nuclear interaction and the formation
of the compound state.
The nucleus, in absorbing the incident beam, changes the intensity
of the outgoing portion of the wave. If there is no absorption, J 77 j = 1
and we can write i\i = e 2is l and  fe  2 =  y out  2 . With absorption,
some of this wave is removed, and we can write  ^ out  2 =  Win  2
— I ^ remove d  2 . But we will regard the wave removed as altered by the
nucleus proper only at the nuclear surface, writing then :
Wout I 2 = I Win  2  f\ ^urf  2 = I Win  2  f\ Win  2 ■ ^^ (47)
I Win I
1 An important paper (W7) is based on an effort to throw the whole burden of
describing the reaction onto the quantity jj. We shall discuss this in Section 6.
Sec. 5C] The Course of Nuclear Reactions 59
with the quantity / representing that fraction of vW which is removed
by the effect of the specific nuclear forces. But we have already shown,
in (35), that we could write  ^ surf j 2 / ^ in  2 = P. With Eqs. (44) in
mind, we get
I *o»t I 2 = I * ia  2 (1  fP) a abs ~ U in  2  I ^ out  2 ~f P \ fc„ J2
(48)
where we can regard f as an expression of the sticking probability £,
discussed above. This would allow us to extend the notion of the stick
ing probability, as some authors have done, even to reactions where the
statistical notions were not strictly applicable, in order to give a kind of
transition between statistical and other views. We then can take as a
fundamental relation for the cross section for compound nucleus for
mation :
*,,= (2Z+l>*. 3 Ptf(^)&i (49)
with the barrier penetrability and sticking probability explicitly indi
cated.
C. The Disintegration of Compound States
1. Competition. The next step in the nuclear reaction, the disinte
gration of the compound state, takes place with a probability D p =
r p/ 2l V (41), as we have already said. We ought to recall here the na
ture of the assumption behind this way of writing the reaction: we
assume that particular features of any single compound level are not
important. Many states are involved, with random phase relations,
either because of the poor definition of the particle energy in the incident
beam, or because the states are so widened as to overlap. Otherwise,
the results we give will be valid only for averages over many states or
even over many similar but not identical nuclear species.
The most striking consequence of the expression (41), for D p is the
phenomenon of competition. Evidently once the compound state is
formed it can decay in many ways, for each of which there is a value of
the appropriate T,. The energy of the compound state will determine
the value of the r/s in our approximation at least. As the energy
varies, given reactants may produce different products. As each new
energy threshold for particle emission is passed, the corresponding r,
rises from zero, and the compound state has then another possible decay
mode. Since the total cross section is limited, the rise of one Tj must
reduce the yield for the others. This is called competition. For a
60 A Survey of Nuclear Reactions [Pt. VI
definite example, consider the reactions induced by neutrons incident on
Br 81 . The reactions which have been observed are four (S5) :
(i) Br 81 (n, 7 )Br 82
(ii) Br 81 («,p)Se 81
(hi) Br 81 (n,a)As 78
(iv) Br 81 (n,2n)Br 80
At very low incident energy (thermal neutrons) only T y is different from
zero, and the first reaction takes place exclusively. As the energy in
creases, the threshold for the second reaction is reached at a couple of
Mev, and it begins to appear. At still higher energies the latter two
reactions are possible, and their competition is noticeable in the cross
section of the (n,p) reaction. It is clear that the computation of <r„ for
the formation of the compound nucleus will apply to all these reactions
(though the resonance neutron capture will need special treatment) and
that the values of the individual cross sections can be obtained if only
the relative T's are known.
2. Specific Level Widths for Particle Emission. We have already de
scribed, in Eq. (33), how the widths are related to cross sections, and
we have given formulas for the computation of individual widths. In
practice, of course, we want not so much the width for emission, say, of
the proton in reaction (i) above, leaving the Se 81 nucleus in a given state,
but rather the total width for the entire range of proton energies and
residual nuclear states which are available. Of course, we have but to
build this up out of the various individual widths, and we shall obtain
at the same time the often interesting energy distribution of the out
going protons.
From (33) we have the value of T pa for the emission of a given particle
with definite energy (averaged over many levels). If now the residual
nucleus can be left in many possible states a, so close that they too can
be represented by the statistical expression for the level density, we ob
tain, for the total width for particle p, T p = ^ T p i a , which we can write
l, a
in favorable cases as an integral:
1 /»Ej, max m fi
—  — • ~j<7 p {E)p R (E™**E p )dE p (50a)
r P =
Here all factors are known: a p is obtained from suitable use of (49),
the level densities are given by some model, taken say from Eqs. (31),
and only the factor £ which occurs in <r p expresses the model's lack of
Sec. 5C] The Course of Nuclear Reactions 61
preciseness. (We suppress the weight g R , including it in the value of PR;
the level density.) We hope to find the £ variation small, and the value
of £ near unity except in special cases. Thus f p can be computed for
charged nuclear products or for neutrons. We shall have to reserve
the gammaray width calculation for a later section; it is almost always
small compared to heavyparticle widths as soon as the emission of
heavy particles is energetically allowed. For the computation of D p
clearly the factor l/ Pc (W) is contained in all terms r y , and the result is
a function of E p maK only. The dimensionless integral of (50a), without
the factor \/t? Pc , is referred to in the literature (W6) as the / P (Ep max )
for a given state (S5). Under specific headings we shall discuss the
calculation of these /'s in more detail:
/ P (^ max ) = ir 2 Pc(W)T p (50b)
3. The Thermal Analogy: Temperature, Cooling, Evaporation. From
the definition of the function f p it is seen that f p is just the width for
emission of particle p from the excited compound nucleus, measured in
terms of the mean spacing of levels at that excitation of the compound
nucleus. There is in addition the numerical factor w 2 . For this reason
we shall refer to the dimensionless / function as the specific width for
emission of particle p.
It is interesting to view the process of emission of particles from the
compound state as a kind of thermal evaporation. Just as a water mole
cule may evaporate from a drop of water, so does one of these nucleons
leave the excited nucleus, in which the excitation energy plays the role
of thermal agitation. Equation (50a) leads to an expression for the
energy distribution of the emitted particles, which is of course just the
integrand in the expression for T p . We have for I(E P ) dE p the relative
number of particles p emitted with energy E p in the interval dE p :
I(E P ) dE p = I E p <t pPr (E™*  E p ) dE p (51)
where the constants have been lumped into 7 . Now from Eq. (30) we
can write the level density PB in terms of the entropy of the nucleus, S.
We get
_ IpEpffp exp [SnjEjT*  E p )]
T(2irdW B /dT) y *
Now we shall make an approximation which is justifiable only for
sufficiently high values of the excitation energy of the residual nucleus
(£p max  E p ). We shall expand the entropy about the maximum value
62
A Survey of Nuclear Reactions
[Pt. VI
of excitation energy which can be left behind in the residual nucleus,
namely about E p ma *. We obtain then
dS
S(W B ) = S(E™*  E) = S(E p ma *)  E p ■
dE p _ r
Now, remembering that the exponential varies much more rapidly
than the factor l/(dWR/dr)^, we use the familiar thermodynamic
relation dS/dE p = 1/r. Inserting this, we get
(IE P ) = const X Ej
p<r p expl I
(52)
Here r is the temperature (in energy units) at which the excitation energy
of the residual nucleus is on the average Wr = E p
If the function
^
a
fc)
\ Neutrons
c
1)
S
N. Protons x 10
<D
>
'+3
"/ s ^^^
£
/ / ^^»^
i ^ ^^^k.
~3
/ ^*r ^^
«
' ^^ i i
' ^>
4 8 12 16
E p , kinetic energy of emitted particle (Mev)
Fig. 6. Energy spectra of evaporated nucleons. Note reduction of proton
evaporation due to Coulomb barrier.
Epff p is not too rapidly varying, as it is not for neutrons of some Mev
energy, for example, the emitted particles have exactly the Maxwell
distribution in energy, just as do the molecules evaporating from a drop.
The temperature T corresponds to the temperature of the drop after
emission of the particles — the temperature of the residual nucleus —
which is not the same as that of the compound state, for of course the
single emitted particle has a nonnegligible fraction of the excitation
energy. This makes the evaporation analogue somewhat less exact;
we must think of a drop with only a hundred water molecules. If the
particles p are charged, the cross section will be quite energysensitive;
low values of E p will mean that the penetrability of the barrier is low,
and will distort the Maxwell distribution, shifting the maximum to high
energies (see Fig. 6).
Sec. 5C] The Course of Nuclear Reactions 63
If the variation of a p with E p is neglected, the maximum energy of
emitted neutrons will lie at E p = r. Table 4 gives temperature as a
function of excitation energy for several nuclei; these temperatures were
obtained by using the level density expression of (31). That they are
clearly rather small compared to the excitation energy helps justify
the approximations made. This means also that the outgoing particles
TABLE 4
Nuclear Temperature in Mev, Atomic Number and Excitation Energy
A = 80 120 160 200 240
Excitation
Energy (Mev)
2
0.61
0.51
0.46
0.43
0.41
5
0.96
0.81
0.73
0.68
0.65
10
1.36
1.15
1.04
0.96
0.91
20
1.93
1.62
1.47
1.36
1.29
will in general take away only a small part of the excitation energy,
leaving behind an excited residual nucleus, which may in turn boil off
still another particle, and so on. It is this process of gradual "cooling"
by "evaporation" which is often the origin of the complicated multiple
processes listed in Appendix II. 1
It is not easy to justify the two assumptions: (a) the statistical treat
ment of the level density, and (b) the neglect of E p compared to E p max .
The level density formulas certainly overestimate the density of the
lowest levels, or at least the lowest ones accessible from a compound
state experimentally defined. Certainly the formula will not be very
reliable below E p m »* of 5 Mev, and for # p max  E p of less than a couple
of Mev. Above that it should be fairly reliable, until the excitation
energy reaches values of 10 Mev or more per particle. The lightest
nuclei will of course make difficulties for the statistical treatment.
The release of neutrons from the moving fission fragments by evapora
tion is to be expected and should account for the prompt fission neutrons,
and a spectacular example of "boiling off" is found in the highenergy
fission of bismuth, where apparently the most probable fission fragments
indicate that the compound nucleus which divides by fission is 12 mass
units lighter than the state initially formed. This implies the loss of
many neutrons by something very like the evaporation we have de
scribed. The nuclear "stars" of cosmic rays often represent such proc
esses. We shall discuss them at greater length in Section 11D1.
1 See, for example, (T5).
64 A Survey of Nuclear Reactions [Pt. VI
SECTION 6. THE DISPERSION THEORY:
RESONANCE REACTIONS
The statistical theory which we have described has been based on an
avoidance of reactions in which the properties of a single compound
state are prominent. But some of the bestknown and most interesting
nuclear reactions are exactly those in which spectacular resonances
indicate the importance of the properties of the individual states.
Starting from the idea that such levels must be considered in detail as
individuals, several authors (B13, B21, K2, S9) have tried to find a
general quantummechanical solution of the manybody problem pre
sented. They based their treatment on the same twostep notion of the
reaction as does the statistical treatment of Bohr. The compound
nucleus is the system of incident particle plus target, a system whose
eigenvalues are not real and whose eigenstates are not stationary but
widened and quasistationary, as we have described them. The forces
of interaction between incoming particle and target cause transitions
between the initial state of the incident particle and an intermediate
state which is one of the widened states of the whole system. Then this
state, by virtue of the internucleon and nucleonradiation perturbing
couplings, itself decays to the state which contains outgoing particle
and residual nucleus. This description was called the "dispersion
theory" because of its formal nearidentity to the calculation of the
optical dispersion of atoms, in which the incident quanta are first
absorbed by the atom, and then another quantum, perhaps the same in
energy as the first, is reemitted by the excited atom. The initial and
final states of the nuclear system, then, are combined only by the mixing
through the compound state, and not with any considerable probability
directly. This is the consequence and of course the motivation of the
idea of the compound nucleus. The "mixing" is strongest when the
energy of the initial system coincides with some more or less welldefined
energy level of the compound state. It is of course the intention of this
form of theory to give a complete account of nuclear reactions, and the
results of statistical theory are expected to follow from dispersion theory
calculations when suitable averages are taken, and suitable assumptions
made about the relative phases of the wave functions involved and
about the character of the levels of the compound state. Whenever
really sharply defined incident energies are experimentally available,
and whenever the states of the compound nucleus are reasonably well
defined, the dispersion theory ought to yield more information than the
statistical model, however improved.
Sec. 6A] The Dispersion Theory: Resonance Reactions 65
The program of the dispersion theory is too ambitious for full success.
The method of perturbation theory which works well for the weak
radiation coupling of the electrons in an atom cannot be expected to
give, even in theory, an adequate scheme for the calculation of the result
of the strong interactions among nucleons.
Wigner and coworkers have given a beautiful general theory (see
Appendix II), if a rather complicated one, which does not employ the
ideas of perturbation theory but insists only that the nuclear forces act
within a welldefined and not too large region of space. Using the ideas
of the ordinary Schrodinger wave equation, we shall indicate the physical
connection between a simple onebody model of nuclear reactions and
the most useful results of the complete dispersion theory, but for full
discussion of Wigner's Smatrix treatment the literature should be
consulted.
A. The OneBody Model and Its Difficulties
The principal notions of nuclear reactions before the early 1930's
were based on a model much simpler than the one we have been dis
cussing. It was built up largely from a study of the decay of alpha
radioactive nuclei and the light nuclear resonance reactions with alpha
particles. The picture was that of the Hartree model of the nucleus, i.e.,
that the many nucleons produced a net potential well, a combined force
field in which the particle to be studied moved. In alphadecay the
alphaparticle shuttled back and forth in this well, until, once in a while,
by chance it leaked out of the barrier by the familiar "tunnel effect" of
quantum mechanics. For scattering, the nuclear forces supplied a
potential well in the same way. To account for the resonances observed,
the charged particle was thought of as penetrating the external barrier
and moving into the potential well; when the particle wavelength was
just such that the particle could produce a standing wave in the well
by virtue of inphase multiple reflections, the particle energy was in
resonance with one of its possible stationary states in the potential well.
Constructive interference built up the wave function strongly, and
absorption grew very rapidly at several welldefined energy levels.
This worked quite well for such reactions as Mg 24 (a,p)Al 27 . It seemed
indeed to be a reasonable model. The first slowneutron resonance
work showed up its major weaknesses. It became evident that in this
theory scattering and absorption cross sections should increase together.
Barring unusual selection rules, the increased particle amplitude inside
the well should lead to increased reemission — scattering — no less than
to increased absorption. Even if capture were somehow prevented for
some scattering resonances by peculiar effects, which might limit the
66
A Survey of Nuclear Reactions
[Pt. VI
dropping down of the particle through radiation to a lower level in the
potential well, it seemed hard to understand why every level which
permitted capture would not yield an even stronger resonance scattering.
The simple analogue of the absorption of sound of the resonant frequency
by a Helmholtz resonator is complete. The constructive interference
of the wave reflected within the resonant cavity builds up its amplitude;
sound energy is not only absorbed, but also strongly reradiated. Yet
experiment showed that strong absorption is not accompanied in general
by strong scattering.
We have already formulated the general problem of scattering and
absorption by a nuclear center of force. Let us inquire into the results
of a onebody model, using this formalism (Eqs. 41 et seq.). Consider
the case appropriate for thermal neutrons, with wavelengths very large
compared to nuclear dimensions. Then we can write for the radial part
of the wave function in the region outside the nuclear radius R
* =
1 u(r)
(ti)^~
«( r ) = V e ikr + e ikr
(53)
Only the 1 = partial wave is important, and outside the radius R there
are no forces. From Fig. 7 we can see that this external wave function
r^ oat = ve ikr +e
V
Fig. 7. Wave function for scattering of thermal neutrons from nuclear potential.
must be joined to the internal solution. For definiteness let us consider
that the internal potential acting within the radius R is represented by
a rectangular well. This is in no way essential for the result, but it
simplifies calculation greatly — at the expense of realism. Within the
nucleus, then, the neutron has the wave function
^int =
u(r)
u(r) = A cos (Kr + e)
(54)
Sec. 6A] The Dispersion Theory: Resonance Reactions 67
with K some wave number, much larger of course than the wave number
* of the slow neutron outside the nucleus, and e some phase constant
The boundary condition at the origin will demand that e = nr/2, with
n odd. The wave equation now requires that the constants be so ad
justed that both ^ and ty/dr remain continuous across the nuclear
boundary. Let us define the quantity /, the logarithmic derivative of
the wave function at the nuclear edge made dimensionless by a multi
plicative constant :
/ = R (du/dr)/u  r=K (55)
Then the continuity conditions are both satisfied if we require / in = f outl
(r,e ikR — P ~ikR\
KR tan (KB + e ) = ikR — i
One %kR + e~ ikR )
Solving for rj, we obtain
2ikR (#in  k R)
Win + kR) ^ 7)
Recall expressions (44). They give
(56)
ffrf » (0) = p d ~ I " I 2 ) and ,/) = J: ( i + ,  3)
k 2
If we find v , both cross sections are determined. Clearly we have not
yet allowed for the possibility of absorption— we have used only one
level within the welland our  „  2 = 1. This follows from the form of
(57) and the fact that / in is purely real. Now we make use of the small
value of kR « 1 for slow neutrons. We can write (57) as
_ (if + kR)(if  kR) fc 2z,2
'= fTm *" d [1 + F ^'f^
by expanding. Then the cross section becomes
<r^=~[l + \v\ 2 + 2ne( v )]=^.Jf^
k 2 k 2 f + k 2 R 2
<kir 1
~ K 2 ' tan 2 (KR + nr/2) + k 2 /K 2 (58)
This cross section shows a maximum at zero energy regardless of the
nuclear size. But, for nuclear sizes and internal motions such that
KR + mr/2 = 0,7r, • •, the small cross section at low energy becomes
instead an infinite peak. If we fix a definite value of slowneutron
energy and imagine that the nuclear radius, say, varies smoothly
68 A Survey of Nuclear Reactions [Pt. VI
strong scattering resonances will appear for special values of the nuclear
radius. Somewhat more familiarlooking resonances, appearing for
definite values of incoming neutron energy, would have resulted had
we considered partial waves of higher angular momentum, but the
general features would not have been very different.
Now, moreover, we can calculate the amplitude of the wave inside the
well, taking an incident wave of unit flux. Using (54) and (56), we get
1 o / n A
A 2 ^
K
Thus the amplitude of the neutron wave inside will become large at
resonance, and in fact we can write the cross section for scattering in
terms of that amplitude, at least near resonance :
o ac (0) = 4xA 2 near resonance
But, clearly, the radiative transitions to a lower state will have a matrix
element proportional to the amplitude A, and the probability of such
transitions, and thus the cross section for absorption, will vary like A 2 ,
as we should physically expect from the fact that A 2 measures the time
the particle spends inside the nucleus. The ratio <x a bs/o S c will therefore
show no marked change at resonance; strong scattering and absorption
resonances will be found at the same energy. This is contrary to all
experience. Furthermore, it is easily seen that the resonance levels
are spaced in energy much too little to correspond to adding another
halfwave to the inside wave function, and that the variation of cross
section with nuclear radius (i.e., with the mass number A) is far too
erratic to be accounted for on such a picture. The onebody model
gives much better results when the external wave function determines
the broad course of events, as in alphaparticle radioactivity. Where
the interior state of the nucleus is decisive, the model is inadequate.
The strong physical plausibility of the compound nucleus picture, added
to the difficulties of the onebody description of neutron capture, gives
it its present importance. The modifications which the strong effects
of shell structure will certainly require, especially at moderate excitation,
have still not been completely worked out.
B. The Dispersion Theory for an Isolated Resonance
The difficulties which surround the derivation of the theory of nuclear
reactions by perturbation methods have led to reconsideration of the
basis of the whole matter. It turns out that the most important features
of the theory arise from the fundamental nature of scattering and ab
sorption processes themselves, which are displayed fully only in the
Sec. 6B] The Dispersion Theory: Resonance Reactions 69
rather complex nuclear domain. The general theory has been developed
by Wigner and several of his coworkers. We shall discuss mainly a
quite satisfactory but much simpler and less general approach developed
by Weisskopf and others. In it the features of the onebody model which
make scattering and absorption so pictorial are generalized to fit the
physical picture of a compound nucleus. In the onebody model we
have shown that, for a case when only two alternatives, elastic scattering
and absorption, are possible, the logarithmic derivative function /
determines the amplitude 77, and hence both cross sections. But in the
onebody model the complex number / (which must be complex to allow
absorption) is fixed by the nuclear potential well. Both modulus and
phase are given. Weisskopf regards this same quantity/, now in general
a complex number, as determined by the whole structure of the nucleus.
It will be different from level to level, varying in a manner much more
complicated than does the onebody /, and in particular having modulus
and phase independently varying. We define the resonance energies
and level widths by giving the properties of/. Thus we have a kind of
phenomenological compound nucleus picture into which the properties
of many levels enter, but through a single function whose determination
from the actual makeup of the compound nucleus we leave perforce
to the physics of the future. We shall require that (1) / is a function
only of the energy (and other constants of the motion) of the compound
nucleus, and not explicitly dependent on the incoming particle, and
(2) / is a welldefined function, defined by (55), in spite of the uncer
tainty of the value of R, the nuclear radius, at which internal and ex
ternal waves are to be fitted. Both of these conditions can be met by
the requirement that the wave function just outside the nucleus vary
only slowly over a distance which corresponds to the mean spacing of
the nucleons within the nuclear matter. We can regard the value of R
as a parameter to be chosen to give the most reasonable average behavior
of /. Some values of R, and the related internal wave number k, will
not work. No value will allow us to assign the behavior of / uniquely,
but the best values will yield reasonable statistical agreement with the
observed properties of many levels.
1. Derivation of the OneLevel Formula, (a) Without External
Forces. Guided by physical considerations, we shall now try a general
ization of the onebody model. The function / is defined by the ex
pression
f = R (du/dr)/u \ r=R (55)
We expect zeros of / when the slope of the wave function vanishes at the
nuclear edge, and infinite values if the wave function itself vanishes
70 A Survey of Nuclear Reactions [Pt. VI
there. Without loss of generality we can take over the special form of /
from the onebody theory:
/= kR tan z(W) (59)
The tangent function no longer has as argument the simple and well
defined phase of the internal wave function in the onebody potential
well, kR + «. It now has instead a function z(W), which increases
monotonically as the excitation energy W = Ej, in< j + Eun increases,
and takes the value nx at each of the successive resonance energies
W r = E b + E T , but may vary as it will in between. The results of the
theory now depend only on the assumption that the variation of z(W)
with energy is as smooth and simple as possible. We shall see how all
the properties of the successive resonances can be described by a suitable
trend for z near each resonance. The prediction of the behavior of z in
detail is given up for the present theory; it could be determined only if
the actual motion of the entire compound nucleus in every eigenstate
were known.
But the onebody picture did not explicitly include absorption. We
allow for that here in the familiar way: we introduce a damping factor
to reduce the amplitude of the now only approximately stationary state.
The damping arises of course out of the possibility that the system can
change its state not simply by decomposing to reemit the incident
particle, but in some other way. Such a damping will as always be
expected to widen and depress the resonance peaks. We write the
energy as a complex quantity: W = W — zT a /2 so that the energy
eigenfunction becomes
Now the probability of occupation of the given state decays in time with
the factor e~ r " t/h \ \[/(0)  2 , and the mean lifetime of the state is given
by r = h/r a while as usual the uncertainty principle will imply that
the state is defined only up to a width r a ~ h/r. This will indeed follow
from the calculated resonance shape.
Evidently the concept of resonance level is useful only if the energy W
has but a small imaginary part, i.e., if T a /W <K 1. Although we could
write the function quite generally, we shall use the approximation of
expanding / in the neighborhood of the resonance energy, leaning
heavily upon the smooth behavior of z(W) and the expected smallness
of T. We write then
_ iT a df I
f(W)=f(W) — = +•••
2 dW\w=w
Sec. 6B] The Dispersion Theory: Resonance Reactions
71
Taking our cue from the onebody model, let us define the resonance
energy, E r , by the relation
f(W r ) = f(E b + E r )=0
and continue to expand / near the value E r . We get
f(W) = (E  E r )
dj_
dE
E r
iT a df
2 dE
+ ■
(60)
E r
where we use the kinetic energy of the incoming particle, E, as measure
of the excitation energy W.
This linearization greatly simplifies our theory. It is certainly no
serious source of error provided that we look in the near neighborhood
of a resonance level. How far that neighborhood extends depends of
course on the variation of the argument z(W) with energy. Now we
can use the relation between the function / and the phase shift r?, just
as we did in the earlier model, from Eq. (57). We take the value of /
near resonance from (60). If we introduce explicitly the functions
fo(E) and g(E), the real and negative imaginary parts of the function /,
we obtain .
f(E) = f (E)  ig(E) = (EE r )8 + ^5 (61)
where we have written 5 for the quantity (df/dE)\ Er , and r o = 2g/5.
Using (57), there follows from (44)
kRg
(62a)
Cabs
(0)
=  (1   ,  2 ) = 4,rX 2
k 2 "~ '"" "" (g + kR) 2 +f 2
We can write <r so (0) similarly. The fact that resonance is marked by
/o = is now plain, and the whole expression gives just the familiar
onelevel BreitWigner formula of dispersion theory; compare (62b).
Now, writing the cross sections out in full, but replacing the functions
/o and g by the more physical widths from (61), and defining a width r„
by the relation r n = —2kR/8, analogously to (61), we have
^abs
(0) _
= xX*
■*■ n* a
m
(62b)
+ (E  E r ) 2
(0) _
= 4ttX 2
= 4tX 2
kR
i(kR + g)  f
2 1 n
+ e lkK sin kR
E E r + (T n + r„)
+ e lkK sin k R
72
A Survey of Nuclear Reactions
[Pt. VI
From the results of (62) we can go much further in the interpretation
of the argument function z(E) of (59). Plainly it goes through multiples
of ir, giving zeros for the logarithmic derivative function / at each
resonance. It is also clear that the contribution to absorption, say, of
any level will fall off on each side of the resonance energy, with a charac
teristic width given by T n + T a = T. But this width is inversely
(re+3)x
(w + 2) 7T
(nhl)T
r large
small
Excitation energy
W
Fig. 8. The argument function z(W) as a function of the excitation energy "FT in a
region containing several resonances, W a , Wb, • • •, with differing widths (F7).
proportional to the slope of the / function near the resonances. From
(57) it is easy to show that
. 2 _ 1 + a 2 Im (/)
1 " ' 1  a 2 Im (/)
where a 2 is positive definite; and, since we require from the absorption
crosssection formula (44) that  17  2 < 1, it follows that Im (/) < 0,
and in our expansion therefore 5 must be nonpositive. This justifies
our use of the minus sign in the definition of T n and r o and confirms
the choice of the tangent function in (59). We can now see graphically
the meaning of Fig. 8, in which the function z(E) is plotted schematically
in a region of several resonances. The width is given in our approxima
tion by T = — const/5 = — const dz/dE. Since 5 is necessarily non
positive, z does increase monotonically, as we expected. Where z changes
rapidly with E near a resonance, 8 is small and the widths large. We
have drawn narrow levels at resonance energies a, b, c, wider ones at
d and e, and a very wide one at /. We cannot predict the level positions
Sec. 6B] The Dispersion Theory: Resonance Reactions 73
or the slope near those values. But we have thrown the whole burden
of the determination of reaction and scattering cross sections into the
behavior of a single function z, which is a kind of equivalent internal
phase. If z varies smoothly, we can expect that dz/dE will be small,
and the widths small when the levels are closely spaced; and reverse
will also hold. By such simplifying assumptions on the smoothness
and statistical regularity we can produce various results concerning the
average behavior of nuclear resonance lines. Individual levels must be
examined empirically, but can be fully described by this theory. The
choice of the matohing radius R can be fixed by the pragmatic test of
how simple and statistically unbiased is the behavior of the many levels,
for R must divide the region outside the nucleus from the one within,
and this division will be physically most clear cut for a particular R.
Improper choice of R will allow the behavior of z to be affected by the
external region and will distort the expected intelligibility of the proper
ties of z.
(6) With Barrier. So far we have considered only the case appropriate
to slowneutron reactions, where the orbital angular momentum I = 0,
and neither centrifugal nor Coulomb forces are present. Outside the
nuclear radius R the potential is strictly zero. In this case, and in this
case only, the external wave function is given not just asymptotically
but everywhere outside the nucleus by the partial plane waves of (53).
It is not very hard to extend the calculation to the more general case.
We write the radial part of the wave function in the external potential
as before, for a definite value I of the orbital angular momentum. But,
as we approach the nucleus, the wave function is no longer a simple
plane wave. We introduce the independent solutions of the wave
equation with the given external potentials which go over asymptotically
into incoming and outgoing partial plane waves. These solutions we
call u t and u for incoming and outgoing parts, respectively. (The
Coulomb potential can be thought of as screened very far away to
avoid the logarithmic term in the asymptotic phase shifts.) We write,
for unit flux,
1 Ui(r) + V U„(r) 1 e i^rl,l2) + + i {k rhl2)
Ur) =W* r — ^ " (63)
Now, as we come in to the matching radius R, the behavior of the solu
tions Ui and u is completely known for any given external potential.
Since they are adjusted asymptotically to the same amplitude, and one
solution could be obtained from the other by simply reversing the direc
tion of time, one is the complex conjugate of the other, u t = u *. We
74 A Survey of Nuclear Reactions [Pt. VI
shall also need the penetrability of the external barrier, which we can
define, as in (35), by the relations
pd) = I "^ I = i_ = i (64)
M,(°0) J 2 _ 1 1_
«,(#)  2 ~ I Ui(R) j 2 ~ Ui(R)u {R)
Here we affix the I value corresponding to the orbital angular momentum
partial wave involved. We shall also make use of the Wronskian rela
tion, obtained in the usual manner by writing the wave equation for
each solution, crossmultiplying, and subtracting, to get
d / du du{\
— I Ui u — ) = uiuj — u Ui = const
dr \ dr dr /
Since this is true for all values of r, we evaluate f or r — * » to obtain
UiU '  u Ui' = 2ik — = 2ikP (l) (65)
U Ui
Now let us define the quantities Fi i0 = RuiJ /ui i0 \ r= u by analogy to
the function /. With all this not very complicated machinery, we have
the solution at hand. Again we write down the matching condition at
the nuclear edge, just as in (56) :
/in = kR(u/ + rfu ')/(ui + r]U )\ r=B
Solving for the amplitude v, we get
Ui{R) Fi  f in Ui {R) (if ia  iA + y)
V = = (66)
U (R) F — /i n U {R) (if in  iA  7)
where we have written the expression in the form closest to (57) and have
introduced the notation F,, = A ± iy for the complex numbers Fi i0 .
Writing /i n = f„ — ig as before, and evaluating y from relation (65),
Fi  F = 2iy = 2ikRP m y = kRP w (67)
we get from the fundamental relation (44) between o„bs and 17, the result :
±irX 2 kRP<> l) g
This is very like the result for the s wave obtained in (62a), and, indeed,
for I = and no external forces, «,,„ = e ±tkB , P (0) = 1, and A = 0,
which gives exactly (62a).
If we write for the unimodular number Ui(R)/u (R) = e~ 2ia , which
was just e~ 2%kR for the noforce case, we can write the scattering cross
Sec. 6B] The Dispersion Theory: Resonance Reactions 75
section as well, and, if we introduce the same linear approximation as the
expansion of (61), we have
<r abs (0 = xX 2 • (21 + 1) _£=!• f68b)
(E ~ E r )* + (r„ + r a ) 2 /2 {bm
crj l) = 4ttX 2 (21+ 1)
in which we have set
+ sin ae ta
T n = 2kRP«)/~
dE
(E  B r ) + (i/2)(r„ + r u )
r« = 2g/,_
«,(•>
(68c)
and defined the resonance energy E r by the relation
(E r  E r «»)8  A = (68d)
E r = E r m +  with/ (#/°>) =0 5 = —
S dE _ r
Equations (68) are the principal results of the onelevel theory, and they
exhibit a number of interesting properties which we shall discuss briefly.
2. Features of the OneLevel Theory, (a) Level Shift. It will be ob
served that the resonance energy is increased by an amount A r = A/5
from the value of the energy E r m at which the value of z was set equal
to nir. Since this value of the energy is in any case not observable
directly, nor calculable in the present theory, the shift might be regarded
as meaningless. It has been pointed out, however, that in the com
parison of the successive levels of mirror nuclei, whose energy levels
might be expected to differ by an easily calculated Coulomb energy,
and in no other way, the effect of A r could be seen, for the excited states
of two mirror nuclei will in general be capable of different modes of
decay, since thresholds for charged particle emission will differ. Thus
A r will differ for the two nuclei, and the observed resonances will not
show energy displacement exactly equal to the Coulomb energy differ
ences. Qualitative agreement with the observations has indeed been
found in at least one case (El), that of the pair N 13 , C 13 . Physically
the level shift can be ascribed to the fact that in a compound level the
particle which ultimately leaves the nucleus spends considerable time
outside the nuclear radius and within the external force field before its
departure; its wave function extends with sizable amplitude beyond the
distance R. A more formal but somehow familiar description is to
observe that such a shift represents the reactive part of that coupled
impedance whose resistive part gives the familiar level broadening and
damping.
76 A Survey of Nuclear Reactions [Pt. VI
(b) Reduced Widths. The particle width r„ is given by (68c) as just
'"""'(ID"
It is easy to compare this with the result of the compound nucleus ap
proach in Eq. (38). We see that here too we can define a "width without
barrier," or reduced width, by writing F n = 2kRP (  l) y r . The magnitude
Y r is just the reduced width, which is dependent only upon the internal
state of the nucleus, as described by the function /. Comparison with
(38) shows that the reduced width is a measure of the probability of the
particle being at the nuclear surface. The absorption width also can be
written in this form, introducing a wave number k to represent some
characteristic wave number for internal nuclear motion. We write the
hitherto unspecified imaginary part of / near resonance in the form
+g = icRha, and then we obtain r also in terms of the reduced width
and of a dimensionless expression giving the imaginary part of /:
r a = 2KRh a y r
The observed width of a level is not always given by these formulas, since
the variation of A in (68a) as the incident particle energy is varied is not
always negligible; this effect can be computed by expanding A r itself
about the resonance energy in (68a), and retaining only the linear
variation. For swave neutrons, of course, A = 0, and there is no such
effect; but cases have been exhibited in which the observed width differed
from the value of r = V„ + r o by as great a factor as 2 or 3.
(c) Negative Peaks. Formula (68b) for the scattering cross section
exhibits some very odd properties, which arise from the possibility of
interference between the resonance term and the second term, called
the potential scattering term. This scattering term arises from the
effect of the welldefined volume in which the nuclear forces act, and
also from the summed influence of all the other levels of the compound
system. There is no rigorous distinction between these two ideas; in
our more formal picture we can think of the potential scattering as com
ing from the fact that between resonances the value of / is such that the
surface wave must be of very small amplitude, much as though the
nuclear surface were the surface of an impenetrable sphere. Taking
the presence of a potential scattering term as our model gives it, then,
let us examine its possible consequences, at least reasonably near a single
resonance level.
It is convenient to observe that the resonance term, with its varying
denominator which gives rise to the familiar witchshaped peaks, can
Sec. 6B] The Dispersion Theory: Resonance Reactions 77
be written in terms of a phase angle 0. With the notation
%Tn r
(E  E T ) + (i/2)(T n + r„) ~ A + i
2(E  E r )
we write
(r„ + r„) (r» + r„)
r r
= r sin Be
A + i cotan + i
which yields the very symmetrical form:
crj l) = ttX 2  (21 + 1) r sin QeT m + sin ae ia  2
If we consider the resonance term alone, the familiar peak is produced
by the variation with energy of the phase angle 0, going from a value
of zero far below the resonance energy, taking a value of w/2 at exact
resonance, and going to x far above the resonance energy. When the
interfering term is considered as well, we may write the cross section
in the following way to exhibit the interference :
<7sc W
* = — zJoi 7 = sm2 a + r2 sm2 6 + 2r sin 6 sin a cos (6 + a)
Here it is clear that, if r = 0, we have pure potential scattering, which is
constant with energy (for not too great changes in energy, we stay in
the neighborhood of one single level) ; if a = 0, we have the pure resonant
peak. The interference is described by the cross term, linear in r. A
somewhat more understandable form of the expression can be obtained
by a not very obvious transformation, using the trigonometric ex
pressions for sums and differences freely, and employing the iden
tity sin x + a sin (x + y) = (1 + a 2 + 2a cos y) M sin (x + e), tan e =
a sin y/{\ + a cos y), which is best derived directly from the indicated
geometrical construction. The transformed expression becomes
r 2 r
o = h (1 — r) sin 2 a [r 2 + 4(1 — r) sin 2 a] 1A
where
+ r[r 2 + 4(1  r) sin 2 a] y * sin 2 (0 + tf>)
2 sin a cos a
tan 24> = ^— (69)
r — 2 sin a
Here it is explicitly seen that the entire energy variation is contained in
a sin 2 term, which recalls the general results of the method of partial
78 A Survey of Nuclear Reactions [Pt. VI
waves. The phase 6 of the resonant contribution always increases by ir
as the energy rises through the resonance. But the value of the potential
scattering phase angle <f> will determine the shape of the observed "peak,"
which will be a normalappearing peak only if <f> = 0, x, and a negative
peak when <j> = x/2, with a dipandpeak combination for intermediate
values. Such "negative resonances" have been observed, and the dip
preceding a peak has been found in a number of cases. The scattering
maximum will be displaced from the energy value E T of the absorption
maximum because of the potential scattering interference. Such dis
placements are typically rather small, and no clear example can be
cited. Use of (69) and related generalized forms to examine the proper
ties of particular levels, especially statistical weights and hence spins, is
frequent.
3. Spins and Statistical Weights. Up to this point we have ignored
the existence of intrinsic spin for the fundamental particles involved in
reaction, and of total angular momenta for the complex systems. We
fix our attention on a single level of the compound nucleus. We shall
assume that such a level has no degeneracies — all accidental ones being
removed by coupling forces of some finite size, 'even if very small,
within the nucleus — except the necessary degeneracy in spatial orienta
tion of the total angular momentum vector J. This implies, in the
absence of external forces, a (2/ + l)fold degeneracy of the compound
state. But the compound nucleus can be formed in many ways. If the
incoming particle has intrinsic angular momentum s, if we consider only
the single orbital angular momentum partial wave I, and if the target
nucleus has initial total angular momentum 7, then the total number of
different ways to form a compound nucleus is (2s + 1)(2£ + 1)(2Z + 1).
Of these only 2/ + 1 will correspond to the given compound level of
angular momentum in question. Thus, for unpolarized beams incident
on unpolarized target nuclei, and with no measurement of the spin of the
resultant particle, we must multiply the crosssection formulas (68) by
the statistical weight factor:
2J+ 1
° J ~ (2*+l)(2*+l)(2J+l) (70)
The absorption cross section for the familiar onelevel case with 1=0
becomes, for example,
2/.+ 1 . r„r a
""abs — ~~~ 77ZZ T 7 ™
(2« + 1)(2I +1) (E E r f + [(r„ + T a )/2] 2
In general, of course, J is not known, and several possibilities exist.
Even for the specially simple case of thermal neutrons, where only
Sec. 6C] The Dispersion Theory: Resonance Reactions 79
I = can contribute, and s = )4, we get the alternatives I + s = J + ,
I — 8 = J _, and the g J± factor is ambivalently J[l ± 1/(27 + 1)].
Sometimes J values can be assigned by study of the crosssection
magnitudes.
C. The Generalized Theory of Dispersion : Many Levels and Many
Decay Modes
We have treated the theory of reactions only in the simplest case.
We have considered only two alternatives: the incoming wave is co
herently scattered, or it is absorbed. In general the theory must take
account of many possible consequences of the formation of the com
pound state. It may be that the energy is not near a single resonance,
but lies between two resonant values; or the widths may be comparable
to the spacing, so that the effect of two resonances may overlap; or
simply that compound states of different J can contribute to the emission
of a single outgoing wave of fixed I, if spins are present. In all these
cases, as in the case where several product particles are energetically
allowed, we have to take into account the various possible courses for
the reaction.
In a series of papers (see Appendix II), Wigner and coworkers have
presented a beautiful generalization of the process here applied in the
onelevel, twoalternative case. The phase shift rj which described the
reaction by the relation <rj 0) = (x/A; 2 )( 1 + v  2 ) is generalized by
introducing a unitary matrix U, such that
^f = ~\(U~l) if \ 2
where i, f index initial and final particles, not only as to type, but also
as to internal state (excitation), spin orientation, and relative orbital
angular momentum. The theory now produces values for the matrix
elements of (U  1), often called the collision or S matrix, between all
the states representing the various alternatives. Unlike the f(E) of
our phenomenological theory, the matrices are given explicitly in terms
of the Hamiltonian describing the interaction of all the nucleons in the
total system. Progress is made, however, only by the demonstration
that much of the behavior of the crosssection formulas can be studied
by knowing precisely only the interactions outside the nuclear radius,
and then replacing a detailed knowledge of the interior by certain bound
ary conditions on the nuclear surface. This is in strict parallel with the
progress of the derivation we have given. Indeed, the more general
method differs mainly by the complete generality into which it has been
80 A Survey of Nuclear Reactions [Pt. VI
cast, and by the somewhat more complete dynamical specification of
the quantities which give the important results. The theory places the
assumptions of our point of view in full sight, and demonstrates that
the chief features of a theory of reactions come from the ability to specify
with more or less definiteness a surface which can divide a region in
which the Hamiltonian is fully known from one where it is not. We
shall not further discuss the more general theory here, but refer instead
to the literature.
Here it is appropriate, however, to indicate one entirely formal scheme
of generalizing our formulas, like (68), to include explicitly the properties
of more than one level. It will be seen that there is full equivalence
between representing the behavior of the cross sections between reso
nances (i.e., where more than one state must be taken into account) (1)
as we have done it, following Weisskopf, by the use of a function f(E)
whose properties, however, cannot be simply given except in the neigh
borhood of one level; or (2) by considering the summed contribution of
a large, or strictly an infinite, number of levels, whose phase relationships
and individual widths and locations can be known only in principle. In
the absence of a detailed solution of the eigenvalue problem of the whole
compound nucleus, no dispersion theory approach which is not statistical
in nature can give useful results except in the neighborhood of a single
level, or at most of a small number nearby, whose properties can be
approximated.
The formal equivalence of the two points of view comes out clearly by
examining (66). We can write the denominator factor in this way:
9(E) = r /T1N *  = £
fUE)  (A  n) j Sj  E
This is an identity where the sum is to be carried out over all the poles «,
of the function g, i.e., the zeros of its denominator. Placing the sum
into (62a) and remembering the linear expansion approximation, we
can write the simpler formula of (62b) as
Cabs — tX
ZdY'iY)^/
E  Ej + \ (iv + IV)
(71)
The virtue of this way of writing the formula is that it exhibits the
possible interference of the contributions of many levels. It is this
interference which must be taken into account to explain angular dis
tribution of reaction products, especially in light nuclei, where broad
levels are the rule. Here phase relations are evidently decisive. But
the existence of constants of the motion, such as angular momentum
Sec. 6D] The Dispersion Theory: Resonance Reactions 81
and parity, will imply selection rules. The mode of formation of the
compound nucleus can influence the phase of the matrix elements which
occur in the (IV) H of the manylevel formulas. In those cases the basic
simplifying assumption of the independent decay of the compound
nucleus will not be valid.
When, as above, the total cross section, integrated over all angles of
product emission, is calculated, any interference terms arising between
compound levels which have different values of the constants of motion,
total angular momentum, or parity drop out. However, for differential
cross sections experimentally given by angular distribution or correlation
measurements, these interferences are decisive. Indeed, every case in
which the angular distribution of some product exhibits asymmetry
with respect to a plane normal to the beam axis must arise out of such
an interference between compound levels of differing parity. The
beststudied example is that of the gammarays from proton capture in
Li 7 near a strong 440kev resonance. No general form of the com
plicated formulas involved is available, 1 though special applications
occur quite completely worked out in the literature. The T r 's are best
represented as matrix elements between the states involved, and close
attention must be paid to the coherent I states in the incident beam, and
to the various combinations of I, s, s z which can give rise to each com
pound level of fixed J r and parity.
D. Statistical Estimates
It is interesting to look at the value of o abs (0) in the case contemplated
by the statistical theory. We think of many levels contributing in a
region of energy AE at E, with AE/E « 1 ; and we examine the average
absorption cross section CT abs giving only average values for the properties
of the individual levels. Formula (62) gives the contribution of each
level. Replacing the quantity (E  E r ) by a variable of integration,
and using the fact that the width V is small compared to the region AE
so that we can treat X as constant over the level, we get, for each level,
<W 0) &E =
27r 2 x 2 r„r a
(r n + r a )
and, for the entire set of N levels in the interval AE,
. £ 2x 2 X 2 IYiy 2T 2 K 2 NT n T a
(Tabs AtL = 2_,
*■ n ~r * a I n T la
1 For an interesting account of the general properties of angular distributions see
(Y3).
82 A Survey of Nuclear Reactions [Pt. VI
We then write for the number of levels its mean value, N = AE/D,
where D is the mean level spacing, and we obtain
... 2jt X r n T a _
<W 0) = — — — r = r a + r„ (72)
It is clear that this expression is the same as the appropriate form of
Eq. (40), with (49):
(statistical) ^o ^ ^ .. .. o ^a „ 25rT n
ffab. *•"■""*" = xX 8 •!).,». Pq« = tX 2
r d
where the sticking probability is now 2irY n /D, which is proportional to
k ~ v at low energies. We have shown how the resonance contributions
of the onelevel dispersion theory sum to the statistical form if D >>> r,
as we expect.
Before we leave the dispersion theory, we shall cite one useful result
which follows more directly from the generalized theory, or from the
perturbation theory than from our phenomenological approach. The
(reduced width) **, y r Vi , is shown in the generalized theory of Wigner (Jl)
to be expressible as an integral over the nuclear surface of the product
of two wave functions, one corresponding to the interior state of the
nucleus, the other referring to the external motion of the product particle.
The reduced width depends explicitly not only on the state of excitation
of the compound nucleus, indexed by r, but also on the nature of the
state of the residual nucleus, and therefore on the quantum state of the
emitted particle. We have neglected the latter dependence, saying
that only the external motion of the emitted particle was important.
If, however, we sum over all the possible states / of the residual nu
cleus, we obtain a limit for y r which cannot be exceeded. This limit
may be approximately evaluated by using the familiar sum rule for the
product of the two matrices, y r / y2 , and regarding the internal wave
function as expressing the fact that the nucleus has a roughly constant
density. The upper limit thus approximated for y r is
r 3 h 2 3
Tr = i^P>  ~2^~I^ MeV ' f ° r nUCle ° n Width (73)
This upper limit sometimes permits the exclusion of certain partial
waves, when the low penetrabilities associated with high I would imply
a y r violating the sum rule limit. The similarity between this limit and
Eq. (39) is evident; from the general scheme it follows that we would
expect widths near the limit for a onebody model, which would diminish
as more and more levels contribute, or more and more particles take
Sec. 7A] Some Typical Nuclear Reactions 83
part in the nuclear motion. In general, our phenomenological theory
would lead to the result that for a smooth enough variation of the func
tion / the value y r /D would be roughly constant (D is the level spacing).
The departure from this result will measure the significance of devia
tions from the statistical theory.
SECTION 7. SOME TYPICAL NUCLEAR REACTIONS
In this part we shall apply the theory of the earlier sections to some
typical nuclear reaction types which it has helped to explain. In each
case a much more detailed account can be obtained in the original
literature; it is the purpose of this section to indicate the method of
using the theory in sufficient detail, and to point out as well its difficulties
and pitfalls, so that the reader may extend the few examples given,
necessarily briefly, here to the whole range of experimental material.
A. Resonance : The Region of Dispersion Theory
Clearly marked nuclear levels, and incident particle beams with well
defined energy, are known in only two types of reactions. The most
important of these (discussed also in Part VII) is the interaction of slow
neutrons with nuclei, leading usually to capture, but often to scattering
or even particle emission. 1 The second type is the class of reactions
using charged particle beams up to a few Mev of energy on target nuclei
from the very lightest up to the region of, say, aluminum. In these
reactions wellmarked levels can often be found (see Figs. 10 and 11),
the wavelengths involved are not small compared to nuclear dimensions,
and the whole approach must be based on using the maximum informa
tion about individual levels.
1. Thermal Neutron Reactions. This large class of reactions is dis
cussed in Part VII, Section 2B2. We shall here discuss it in sufficient
detail to illustrate the use of the theory presented in Part VII.
In the thermal region, the neutron wavelength is very large compared
to nuclear dimensions. The neutron de Broglie wavelength X = h/mv
is just X = 0.045 A/(E) V2 , where E is the kinetic energy in electron
volts. For energies from a few millivolts up to, say, a thousand volts,
which broadly defines the region of interest in these experiments, X
ranges from 2 A to 10 3 A. Throughout this range, the partial wave
corresponding to I = 0, the socalled S wave, alone will be effective in
reaching the nuclear surface. The penetrability of all other partial
waves is so greatly reduced by the centrifugal barrier that they can be
1 See, for example, the collection of results and references in (G6).
84 A Survey of Nuclear Reactions [Pt. VI
neglected. From this follows the isotropic distribution of scattered
thermal neutrons and of capture gammarays in the centerofmass
system.
(a) Effects of Target Motion and Binding. It is well to introduce here
a note of caution. The usual neglect of the chemical forces acting on
the target nucleus, of the thermal or zero^point motion of the target
nucleus, and of the possible coherent scattering from the neighboring
nuclei of the target material cannot be justified in the thermal neutron
reactions. The coherent scattering of neighboring nuclei (discussed in
Section 5 of Part VII) will show up strongly in crosssection measure
ments as long as the neutron wavelength is near the values which fulfill
the Bragg condition rik — d sin 6 for the lattice spacing d of the crystal
or microcrystals involved. This means that, in the range above 0.1 ev,
the effect is not important for most target materials. The molecular
binding effect is very important in determining the energy loss upon
elastic collision, but again does not affect cross sections much in the
region where the energy of the neutron is greater than the smallest
vibrational level difference in the target molecule or crystal; and in
addition this effect is of small importance for reasonably heavy nuclei.
We shall treat here briefly the effect of the thermal motion of the target
nucleus — the socalled Doppler effect.
Let the velocity of the neutron be v in the laboratory system, and
the component of the nuclear velocity toward the neutron beam be V.
Then the relative velocity of neutron and nucleus is (v + V), and the
relative kinetic energy
fYh
#rei =  (v + V) 2 = E n + (2mE n ) y *V (74)
Jt
to the first order in V/v. Here m is the neutron mass, E n the neutron
kinetic energy. If the target atoms were those of a gas, they would
move with the Maxwell distribution, giving for the fraction / of atoms
having the velocity component V, with atomic mass A,
"■ v > dv '(^rT e ~ m '" ir (75 >
Inserting expression (74), we find, for the probability of a given E ie \,
f{E iei ) dE tel = — • et*™ "> — D = 2 [~~j) (76)
where D is the "Doppler width." The cross section is given by the one
Sec. 7A] Some Typical Nuclear Reactions 85
level formula written in the form appropriate for neutron capture (or
scattering), near the resonance energy, E r :
Un
<r(E„) = (77)
1 + [(E n  E r )/(T/2)f K " }
and E n here is of course to be replaced by the relative energy i? rel .
The term {2mE n ) V2 V is the correction for the motion of the center of
mass. With this we obtain, for the effective cross section S(E n ),
5{E n ) = f*(E Tel )f(E Tel ) dE rel = c F (' x)
with x = (E n — E r )/(T/2), the deviation from resonance in units of
the total width, and the integral
\D / 2(tt)^Z>J_
e rHx X ') 2 /iD 2 ^ x '_
1+x' 2
This function simplifies for large natural width to the form of (77)
exactly as without temperature motion. But, for small natural width
compared to Doppler width, the cross section at resonance is changed
and the shape of the curve altered. In this case, with T/D <K 1, we
obtain simple expressions for both :
(1) E very near the resonance energy, (E n — E r )/D « D/T, when
(*•)* r
_ . 6 (je„fi.)7.D s
2 D
(2) E very far from resonance, (E n  E r )/D » D/T, and
1
^ 1 + [(E n  E r )/(T/2)] 2
the value without Doppler motion.
For E n = E r , exact resonance, we can easily obtain
\D / 2 D \ (tt) 1a J )
If T/D 3> 1, this reduces to F = 1, and the cross section at resonance for
large natural width is just a(E r ) = <r . For exact resonance, and small
natural width T/D « 1, F(T/D, 0) = (r) K /2 • T/D, and the measured
cross section at resonance becomes
(»•)* r
86 A Survey of Nuclear Reactions [Pt. VI
much reduced by the Doppler broadenings. Lamb (LI) has shown that
these same formulas hold even if the target — as is usually the case — is
not a perfect gas but a Debye solid (other models will give similar
results). In fact, if we replace D in (76) byanewvalueD = 2(£ , „/cre q ) M ,
where T e(l is the equivalent temperature corresponding to the mean
energy per vibrational degree of freedom (i.e., kT e<1 = mean vib. en
ergy/vib. degree of freedom), then the formulas are unchanged. This
holds as long as either the natural or the Doppler width is large com
pared to the Debye temperature. For a not too sophisticated application
of the Debye theory, the equivalent temperature is simply related to the
Debye temperature. A few values are shown in Table 5. For other
TABLE 5
Equivalent Temperatubes or Cbystal Vibrations
T eQ /T
2 /"Debye
1
2
1.06
1
1.15
0.75
1.35
0.5
1.8
0.25
cases, consult Lamb and the other references. It will be observed that
all this discussion applies only in the case where the wavelength is such
that neither crystalline nor molecular diffraction effects are appreciable.
(6) Level Widths and Positions. With such conditions in mind, we
can look into the BreitWigner onelevel formula for neutron resonance
absorption or scattering. From (62) or (70) we obtain
aX 2 / 1 \ a(E) 1A \T a forabs
°abs,so = —  I 1 ± — — — 7 ) — — TS : TTS X
2 \ ± 2/ + 1/ {E  E r ) 2 + (r/2) 3 " [a(E) « for sc
(78)
Here we have written for the neutron width, r„, the value r„ = a(E) 1A ,
and we assume that the radiation width r a is constant. These two
simplifications follow from the most marked feature of slowneutron
work: the fact that we are here studying, with resolution of a fraction
of an electron volt, a very small portion of the level spectrum of the
compound nucleus, some 5 to 8 Mev excited. We can safely regard all
factors in the widths which depend on the compound nucleus as constant
over the whole resonance, and from the formula we have the results
employed.
A few thermal neutron reactions are exothermic for heavy particle
emission, as for example those with neutrons on H 3 , Li 6 , B 10 ; N 14 . Here
Sec. 7A]
Some Typical Nuclear Reactions
87
the widths are very large on the scale of thermal energies, many kilovolts
at least, and the cross sections become simply <r a b s ~ X 2 (E) y * ~ 1/(E) lyi .
This is the famous 1/v law. Even in heavier nuclei where gammaemis
sion dominates, the 1/v behavior appears for a range in neutron energy
small compared to the energy of the lowestlying resonance. Only
where the resonance lies very near zero energy (either above it as in
Gd 157 or below it as in, say, Hg) does the 1/v law fail at low energies
(apart from crystal and chemical binding effects, of course).
A great body of experimental information has been obtained (and
compiled in useful form by Adair (Al)) on the shape of the transmission
curve and from it the cross sections for very many isotopes. These
yield on analysis (for the narrow isolated levels for middle to high A)
fairly reliable values for the three most characteristic parameters: the
resonance energy E r ; the total width T and the neutron width evaluated
at 1 ev energy; a, taken from the measured cross section at exact reso
nance, c(E r ). We tabulate a few typical values in Table 6.
TABLE 6
Some SlowNeutron Resonances and Their Properties
Nu
E r
r
aX 10 3
CO
Remarks
Refer
cleus
(ev)
(ev)
(ev)*
(barns)
ences
Na 23
3,000
~170
r« < r„/ioo
550
sc
1
Mn 55
300
~10
~4,000
sc
2
Co 69
115
25
~12,000
sc
2, 3
Rh 103
1.21 ±0.02
0.21
0.5
2,700
abs
4
Cd 113
0.176 ±0.001
0.115
2.2
58,000
abs
5
In 116
1.44 ±0.03
0.085
2.2
27,600
abs
6
Sm 149
0.096
0.074
2.0
110,000
abs
7
Eu 163
0.54
0.15
1.7
20,000
abs
8
Gd 167
0.028
0.12
4.5
290.000
abs
9
Au 197
4.8
<1
>3,000
abs (unre
solved)
10
1. C. T. Hibdon el al., Phys. Rev., 77, 730 (L), (1950).
2. F. Seidl, Phys. Rev., 75, 1508 (1949).
3. C. Hibdon and C. Muehlhause, Phys. Rev., 76, 100 (1949).
4. R. Meijer, Phys. Rev., 75, 773 (1949).
5. L. Rainwater el al,, Phys. Rev., 71, 65 (1947).
6. B. McDaniel, Phys. Rev., 70, 832 (1946).
7. W. J. Sturm, Phys. Rev., 71, 757 (1947).
8. L. B. Borst et al., Phys. Rev., 70, 557 (1946).
9. T. Brill and H. Lichtenberger, Phys. Rev., 72, 585 (1947).
10. W. Havens el al, Phys. Rev., 75, 165 (1947).
88 A Survey of Nuclear Reactions [Pt. VI
It is seen that for all good absorbers the width r is indeed much
greater than the neutron width, a(E) 1A . It is moreover reasonably
constant among nuclei. This reflects the fact that it is made up of a
sum of partial transition widths for the emission of gamma to all the
possible lower states of the compound nucleus, which tends to average
out fluctuations. But recent measurements on capture gammarays do
appear to demonstrate a greater individuality among capture gamma
ray spectra than this point of view would lead one to expect; if those
results are correct, we are seeing again a reflection of the special features
of nuclear level spectra, perhaps arising from the importance of shell
structure.
A few examples of resonance scattering have been observed in some
detail, both for slow neutrons, with I = 0, and for faster ones. The
interference between potential and resonance scattering is plainly seen,
along with the expected dip in the crosssection curves. Most thermal
scattering, however, is simply scattering far from resonance — that
contributed by the potential scattering term. The cross sections ob
served fluctuate more or less widely about the value 4x.ffi 2 , where R is
the nuclear radius. The nuclear boundary cannot of course be very
sharply defined for this process. Nearby levels will cause deviations
which arise from the interference of other resonance terms. For the
lighter elements, below A about 100, the level spacing is large, and the
possible neutron widths therefore also larger (see Fig. 8) ; we may expect
some sizable resonance scattering effects to show up, even rather far
from resonance and without sign of much capture. The actual resonance
scattering observed in Co and Mn is then complemented by some rather
large deviations from the expected value, without evidence of actual
resonance, in Cu, Ni, Fe, and a few other nuclei. Mn itself shows an
abnormally low value of tr sc at thermal energy (H9, S6). 1
The magnitude of the scattering cross section cr sc is measured of course
by transmission. Recently experiments which make ingenious use of
molecular and crystal coherent scattering effects have been applied to
observe the phase change on scattering. These results demonstrated
that most heavy nuclei showed a change in phase of w between incident
and scattered wave (F4). This would be expected from the model we
have used, for, wherever the "potential scattering" term was the most
important, the phase change ought to be just that produced by an
impenetrable sphere. The extension of the ideas of the onebody model
made by the method of Weisskopf et al. still leaves unchanged many of
the simple conclusions from that picture!
1 This whole treatment follows closely that of (F7).
Sec. 7A] Some Typical Nuclear Reactions 89
The high distribution of high capture cross sections through the
periodic table reflects the notion that the presence of a resonance level
just at the excitation energy of the compound nucleus is a matter of
sheer chance, since we have magnified a single halfvolt region out of
many millions. This consideration allows us to ascribe the fluctuations —
which are indeed wild — to simple chance positioning of levels, and of the
general trend to the gradual increase of level density. The cross sections
for capture slowly rise, with fluctuations of course, until in the rare
earths and beyond high cross sections are common. The sharp decline in
the cross section of lead and bismuth is attributed to a decrease in
neutron binding energy, also implied by the general kink in the mass
defect curve which is responsible for the natural radioactive elements.
The level density, which ought to increase as the particle number in
creases for a given excitation energy, actually declines, partly because
of the smaller excitation energy yielded by neutron capture, and partly
because of the influence of closed shells. This is another reflection of
the presence of structural detail superimposed on the statistical behavior
of the nuclear drop; compare Fig. 17.
2. Charged Particle Reactions on Light Nuclei, at Moderate Energy.
The special ease of defining zero kinetic energy, which makes slow neu
trons so nearly monoenergetic, cannot apply to charged particle reactions.
Slow protons are not very hard to make — hot atomic hydrogen — but
they obviously will penetrate the Coulomb barrier so little as to make
them useless for study of nuclear reactions until the temperature reaches
that of stellar interiors, ~10 7 °K, where indeed thermal protons are of
the utmost importance in inducing nuclear reactions, and are the
agents of the release of stellar energy. But beams of charged particles
up to a few Mev energy can be produced which are welldefined in
energy and direction. Indeed the latest techniques * allow an overall
resolution, including the effects of the slowing of the protons in the
material of the target, of the order of 100 ev out of a couple of Mev.
For nuclei and excitation energies in a range where charged particles
of this energy will penetrate the barrier, and where the level spacings
are large compared to such a figure, we may expect to apply the dis
persion formalism to the computation of excitation functions.
It will be clear that the general trend of all such reactions, which are
primarily those with protons and alphas (or any charged projectile)
on nuclei up to A ~ 50, will be governed by the effects of the Coulomb
barrier. This will determine the width for entry T p and for reemission
of any charged particle. But, superimposed upon this easily understood
1 See the work of the Wisconsin group as described in (H7), for example.
90 A Survey of Nuclear Reactions [Pt. VI
extranuclear factor will be the effects of strong resonances, which give
the complex results seen in Figs. 10 and 11. We want to discuss here
the method of treating such reactions; it is already obvious that only
detailed attention to the properties of individual levels will be adequate,
(a) The Reaction Li 7 + p. Figures 9 and 10 show the simplified
experimental results for the two sets of products 2a and (7 + p')' —
inelastic scattering — from the same reactants Li 7 + p. 1 The smooth
rise in Fig. 9 is indeed just a penetrability curve, following the formula
with a width without barrier, G, large compared to the energy variation
for the protons, and an alphaparticle width which is independent of
energy, since the reaction is exothermic by some 17 Mev and the alphas
are far above the barrier. But the rise is not that appropriate for a pure
Coulomb barrier, swave particles alone coming in without angular
momentum; it is necessary to assume that the incoming particles pass
as well through a centrifugal barrier, with 1=1. The »Swave pene
trability would lead to a curve whose rise was nearly complete after
about 500 kev; even more striking, the cross section observed is about
fifty times smaller than such similar reactions as Li 6 (p,a)He 3 . This
points again to the absence of swave particles, which experience a
much smaller barrier. We must assume that such particles cannot
induce the reaction from the compound state of Be 8 involved. The
reaction is governed by a strong selection rule: the conservation of
parity. The two identical alphas emitted obey Bose statistics; they
must have a wave function invariant under their exchange. But to
exchange two alphas is equivalent to reflecting their wave function in
the origin, since they have no spin. The alpha wave function must
then have even parity. Their relative orbital motion will have a parity
of (—1)', where I is the orbital angular momentum. We have then
shown that the parity of the system is even in the final state. Since
parity is conserved, it must also have been even in the initial state.
But all simple nuclear models agree in giving the ground state of Li 7
odd parity. The incoming proton must therefore have had odd parity
if the two alphaparticles are to be formed. This requires that the
wave function for orbital motion of the incoming proton have an I such
that $ ~ Yi m gives odd parity. The values I = 1, 3, ■ • • will do this; of
these, the most easily penetrating partial wave is that with 1=1, the
pwave protons we observe. Thus we can account for the excitation
'See (HI 1) and(H12).
Sec. 7A]
Some Typical Nuclear Reactions
91
o
12 3
Proton energy (Mev)
Pig. 9. Relative cross section as a function of incident proton energy for the reaction
Li\p,a)a.
eg
E
C3 6
bO
S3
o
+3 .
o 4
^5
13
0.25 0.5
Proton energy (Mev)
0.75
Fig. 10. Relative cross section as a function of incident proton energy, for the
production of gammarays by the reaction Li 7 (p,p')Li 7 * in the region of the major
sharp resonance at 0.44 Mev.
92 A Survey of Nuclear Reactions [Pt. VI
curve and the low yield by assuming even parity for the compound
state involved.
The angular momentum of the compound state must be the resultant
of the incoming proton spin, S = J^i an d the target nuclear J = %,
combined with the orbital angular momentum 1=1. This means that
the total angular momentum of the compound nucleus Jc must be
Jc = 0, 1, 2, or 3. But, if the state is to emit alphas, which have no
spin, the angular momentum must be even (from the Bose statistics)
and the state is then described by the quantum numbers J = or 2,
even parity. The angular distribution of the alphas with respect to the
fixed direction of the incoming beam can be no more complicated than
the partial wave of the incoming beam, and in fact the wave function
will be of the form if/ a ~ a + bYi°. Since Bose statistics requires that
the wave function of the outgoing alphas be even in cos 6, the most
general form possible is a + b(E) cos 2 6. This fits the experiment
reasonably well up to some hundreds of kilovolts. Lately the higher
energy angular distribution has shown (H8) terms of the kind cos 4 6,
which imply that incoming /wave particles must be considered. The
general features of the discussion are unchanged.
But what of Fig. 10? The very sharp resonances for the emission of
a 17Mev gammaray must be explained. The cross section for this
reaction at resonance energy, with I the spin of the target nucleus, is
7rX 3 rj,r 7 27 + 1
<Ty(E r ) ~ ■
(r p + r T ) 2 2
and, since the measured value of <r y (E r ) is 7.2 millibarns, with a resonant
energy of 440 kev and a measured total width of 12 kev (F9, F10), we
can write the value of T y as
ffyl abs
r T ~
(2/+l),rX 2 i
assuming only that T y /V p « 1. This yields a gammaray width for
the excited state of the compound nucleus Be 8 of about 30 ev. Ob
viously a strong selection rule must be in play to prevent the decay of
the compound state by alphaemission, which would be expected to
have a width of hundreds of kev. The possibilities are two: (1) Only
swave protons are captured to form this level, giving it angular mo
mentum Jc = 1 or 2, odd parity, or (2) the same pwave protons which
give the previous reaction are captured, giving states with even parity,
but with spins J c = 1 or 3. In either case, whether the compound state
in question has odd parity or odd J c , it cannot emit two alphas. To
Sec. 7A] Some Typical Nuclear Reactions 93
distinguish between the alternative, as to fix the J c in the two alpha
emitting states, requires further information.
This has been provided by a beautiful experiment (D3) in which
accurate measurements were made of the angular distribution of the
gammarays. The experiments fully confirmed the dispersion formula
in the interference form, Eq. (71), by the observed variation of the
angular distribution as the proton energy was varied through resonance.
Forwardtobackward asymmetry proves that two levels are involved in
the gammaemission, with opposite parities. It is natural to identify
the wide interfering state with some state of the (p,a) reaction, neces
sarily even, and then the narrow resonant state must be odd, formed
by swave capture. This is consistent with the observed isotropy of
the gammarays near the sharp resonance. Two lower states take part
in the gammaemission: the ground state, transitions to which yield the
famous 17.6Mev gammaray; and a state at about 3 Mev excitation,
nearly 2 Mev wide. Both these lowest states decay by alphaemission.
The general rule about the zero spin of eveneven ground states would
lead to the conclusion that the ground state here has J = 0. Some
experimental evidence for a lifetime of this state near 10 — u sec does
not seem strong enough to upset the rule, and evidence from the beta
decay of Li 8 (that the two lowest Be 8 states differ in J) supports the
rule. The nearequality of the gammaray transition rates from the
resonant state to the two low states suggests that they do not differ by
many units in J. It seems likely that the 3Mev state has J — 2,
parity even. Then the choice between the two possibilities J e = 1 and
J c = 2 for the sharp state seems direct; we assign the resonant state
the quantum numbers J c = 1, odd, and all the effects are described;
both gammarays are electric dipole, thus fitting with the sizable width.
This neat account seems disproved by observations (C9) of the elastic
scattering of protons at the 440kev energy. The presence of a scatter
ing resonance whose interference with the Coulomb scattering has been
studied shows that at least most of the compound nuclei are formed by
pwave, and not by swave, capture. Then the isotropy of the resonant
gammarays is in the nature of an accident, (C4) implying not that the
protons brought in no evidence of direction, but that the compound
state with J = 1 was formed with equal probability in the three sub
states J z = ±1, 0. This would give isotropy; it will take place, accord
ing to the rules for combination of angular momentum matrices, if the
triplet proton wave (spin parallel to orbital angular momentum) is cap
tured about onefifth as frequently as the quintet wave (D4). Then
the interfering state is some nonresonant odd state, and the sharp
resonant state must be assigned the description J c = 1, even. The
94 A Survey of Nuclear Reactions [Pt. VI
gammarays are magnetic dipole and electric quadripole in a particular
mixture. This is a less simple and less satisfying story, but it seems
required by experiment.
This account is a rather sketchy summary of one reaction, given as
an example of the manysidedness of the problem of nuclear reaction
spectroscopy in the region where individual levels are of importance.
The example at hand is not the simplest known case, but it is very far
from the most complicated. The discussion of each reaction in great
detail will be the content of a final nuclear spectroscopy.
(6) Some Other Examples. Proton capture by C 12 and C 13 has been
much studied. Here the fit to the onelevel dispersion formula is excel
lent, though one must take into account the variations in T p coming
from the appreciable change in penetrability over the 40kev width of
the resonance, which is some 2 Mev below the top of the barrier. It is
worth while to note that the shape of the 440 kev resonance in the
Li 7 (p,y) reaction does not fit the BreitWigner formula especially well.
There is a superimposed asymmetry, with a rising gammaray yield
on the highenergy side of the resonance. But in order to ascribe this
to a variation in penetrability — arising from the capture of a proton
partial wave with high I — a very high centrifugal barrier would have to
be invoked (B19), and this would make the cross section absurdly small.
Some more complex level scheme must be employed.
The complicated and beautiful studies of the reactions arising from
F 19 + p have led to repeated analysis from the point of view of the one
level formula (Fig. 11). Here the problem is to see if all the fifteen
or more levels observed can be fitted into a scheme which gives reduced
widths y r for the various reactions, each roughly independent of the
level involved, and tries to account for the great experimental variation
in width and strength of levels on the basis of penetrability changes due
to* energy and angular momentum values alone. The situation is still
not entirely unraveled, but the scheme is probably workable without
too much arbitrariness (S2).
There is increasing evidence of the value of another approximate
constant of the motion, the socalled "isotopic spin" (W8). This
quantity describes the spatial symmetry of the nuclear wave function
under the interchange of any two nucleons. It can only be approxi
mately conserved, of course, since in fact the neutron and proton differ
at least by Coulomb effects. Isotopic spin would be an accurate quan
tum number, and the nuclear wave function would consist of functions
with one welldefined isotopic spin value, only if all nucleonnucleon
interactions were identical, independent of the type of nucleons involved.
How strong this partial conservation law really is has not yet been estab
Sec. 7B]
Some Typical Nuclear Reactions
95
lished. A similar situation exists for the spin, which can be intercon
verted with orbital angular momentum only in the presence of non
central forces. Strong forces of this type exist, and hence spin is in
general not even approximately conserved; only total angular momen
tum, a strict quantum number, remains as a good basis for angular
momentum selection rules.
Whether the spectroscopy of nuclear levels based on the dispersion
theory can be carried beyond these light nuclei is dubious. Even here
A i ks^J
0.4 0.8
Proton energy (Mev)
Fig. 11. Relative gammaray yield (schematized) as a function of incident proton
energy for the reactions coming from F 19 + p in the region below 1 Mev incident
energy. Note the many and varied resonance peaks.
the level schemes which have been proposed are highly complex. With
out an a 'priori guide to the position and character of nuclear levels, the
job of spectrum analysis is difficult. It will be noticed that the problem
of the light nuclear reactions is quite parallel to the problem of the low
lying states revealed in the gammadecay schemes of radioactive nuclei.
In the present example, however, the natural widths of levels is directly
measured, while the number of possible reaction products adds the
complicating factor of competition. But there seems little reason to
doubt that the general scheme proposed is adequate.
B. Reactions without Marked Resonance
We have seen in Section 5 how the statistical theory of reactions
proceeds by looking apart from the details of individual levels, and in
Section 7A we have discussed in detail some reaction types which are
clearly suited to the description of dispersion theory, in which the proper
96 A Survey of Nuclear Reactions [Pt. VI
ties of single levels are made the basis for understanding. Here we shall
apply the ideas of the statistical theory to the large class of reactions
in which no marked properties of levels are observed; the theory is
intrinsically suited to such reactions. We remark first that these reac
tions are generally characterized by high excitation of the compound
nucleus. Wherever this occurs, the level spacing decreases and widths
increase, as we have seen, so that either for all circumstances or for the
conditions of nearly all practical experiments the importance of indi
vidual resonances is negligible.
Excitation of nuclei by neutrons means the formation of a compound
nucleus with energy of excitation equal to the neutron binding energy at
least. As shown in Section 7A, this region (say 8 Mev in nuclei from
A ~ 100 to about 200, and perhaps 6 Mev or less thereafter) is marked
by discrete levels, spaced from a few hundred to a few electron volts,
when the incoming neutron has negligible kinetic energy. As the
kinetic energy of the neutron beam increases through the region of
marked resonances, the level density increases rapidly. At a few kilo
volts at most, the present technique does not permit resolution of indi
vidual levels. We have already shown [e.g., in (72)] that the average
over many levels of the dispersion formula leads to a result agreeing
with the statistical theory. It is of interest to apply this idea to the
actual case of the absorption of neutrons with energy in the range from
a few to a few thousand electron volts, with the middle and heavy
nuclei as targets.
If we stick to swave neutrons only, we can write for the average
absorption cross section, averaged over many levels, with level density p
(0) = 2xp ■ xX 2 — — ; — — (bars indicate average value (79)
over manv levels')
J n± a
* ' a over many levels)
[Compare (71).] From the expansion for f(E) given in (60) we can write
r„ = — 2kR/8, with 8 and R constants which can be given a rough
interpretation as determining properties of the function z{E) [see
(62)ff.] or can be regarded as empirically determined. We find that
r a y> F„ for energies below a few kilovolts, and then in this region we
would expect
co, ^ R !^
" a ~ k ' s
where R, p, a, 8 can be roughly estimated. Note that this applies to
averages over many levels, not to thermal cross sections. Although
there is no direct check yet on this prediction, it seems consistent with
Sec. 7B] Some Typical Nuclear Reactions 97
the general knowledge of absorption cross sections. A more detailed
extension of this similar theory has given results 1 which are not wholly
in accord with the data, at higher energies especially, but which seem
to demonstrate the essential correctness of the approach, granting
its rather high degree of arbitrariness in fitting the results of single
experiments.
The scattering cross section can be treated in the same way. Here
of course the potential scattering term e* s sin S is important. Even for
very simple assumptions about level widths, a rather complicated result
is obtained which is not yet wholly confirmed. Both at thermal and at
higher energies the importance of the "impenetrable sphere" effects of
the potential term are evident in the total cross sections where no strong
absorption resonance is present. Variations of radius R as well as special
structural features of some nuclei seem to be present in the detailed
comparison of theory with experiment. 1 Agreement is much better for
some heavy and for some lighter nuclei; a few middleweight nuclei do
not give even the expected energy dependence of <r„ (0) , i.e., 1/E* A .
1. The Statistical Approach. For sufficiently high energy of the bom
barding particle, no resonances will be observed. This can arise out of
the experimental conditions : it is hard to define the beam to a very small
energy range as its mean energy grows, and even the temperature motion
of the target nuclei will produce a comparatively large uncertainty in
relative energy for high bombardment energies, as we can see from
relation (74). More significant, however, is the fact that the levels
themselves become broader as the energy of excitation increases. This
is the obvious consequence of the fact that higher excitation energy
quite generally makes available many more modes of decay from a
given level. In this region of high energy, then, the dispersion theory
will have little value, for many levels, each with its unknown but specific
properties, will take part in every reaction.
In the preceding section we showed how the dispersion theory treat
ment went smoothly over into an entirely statistical form when the level
spacing was small compared to the level width, for the particular case of
neutron absorption. We replaced the precise values of the level widths
by averages over many levels; such averages ought to vary smoothly
with energy. In a sense we here calculate not the result of a particular
reaction with a definite target nucleus, but a kind of average over many
nuclear species very close in A and Z to the actual target. This ought
to give then the general course of any reaction, fluctuations about the
average behavior which will show up in particular cases being disre
1 ThisTvhole treatment follows closely that of (F7).
98 A Survey of Nuclear Reactions [Pt. VI
garded. The recent neutron scattering work of the Wisconsin group
seems to show such fluctuations (B3).
The fundamentals of the statistical theory were given in Section 4;
here we propose to apply them. The main relations are two in number.
The first, which is given in Eqs. (40) and (44), is the statement of the
main Bohr idea of the nuclear reaction progressing in two independent
steps:
<r iiP = Sri ■ JS? = xX* 2 E W + DPiiia ■ ■=£ = *i(fi) ■ ~~ (80)
Here we have introduced the cross section <ri(c) for formation of the
compound nucleus with incident particle i. The second is the statistical
relation between level width for a given disintegration and cross section
for the process inverse to the disintegration [see (33) and (34a)] :
r ia (W) Pc (W) = ^ ~ (81a)
and
Y nl = kGP n ,i (81b)
From these two we can find either the width or the cross section, using
basic ideas, and compute the other from their relations.
(a) Neutron Reactions. Here we have only the centrifugal barrier
opposing the contact of neutron and target nucleus. Let us consider
first the case of highenergy neutrons, with energies so high that the
wavelengths corresponding are small compared to nuclear dimensions.
In such a case we may follow the classical trajectory of the incoming
particles. They will form the compound nucleus with sticking probabil
ity whenever they strike the nuclear disk. Moreover, it is evident from
our whole nuclear model that for energies above, say, a few Mev the
value of in is essentially unity: every particle that touches the nuclear
matter sticks. Now we may write the contact cross section for neutrons
in the partial wave of orbital angular momentum I as we did in Eq. (49) :
tr n ,l = (21 + l)*K 2 Pn,itn.i (82)
But for high energies we set £ n i * 1, and the quantities P n ,i [found in
(36)] take the values P n ,i — > 1 for low I's, such that the centrifugal
barrier is well below the energy available, say for I below l c , the critical
angular momentum. For I > l c , the values P„,; rapidly approach zero,
and, if many Z's are involved, we can neglect the transitional cases near l c .
Sec. 7B] Some Typical Nuclear Reactions 99
Then just as in (44b) the cross section <r n (c) for formation of the com
pound can be written
C n (c) = 2(TiZ
l c ~hR
= ttX 2 23 (2* + l)P,m » ttX 2 E (2Z + l)P Bl!
2=0 o
^irP 2 (high energy) (83a)
which is of course the classical value.
At low energies, we can give no a 'priori guess about the quantity
£„,i. But relations (81) for level width can be applied. We write
Tm = kyP ntl = ^ . f=I = i* (2Z + l)P n fa (84)
2:rp irX 2irp
It now seems inviting to make the identification :
2irp7
£„,* = * (85)
gn
Here the functions p and 7 depend only on the compound nucleus
[while gR is the statistical factor needed in <r n i, by (69)], and will vary
very little for small changes in neutron kinetic energy. We write this:
k
£» = 
and
k
<r n (c) = ttX 2 2(2Z + l)P ni! •  (low energy) . (83b)
K.
with the constant K to include the unspecified functions above. The
result is then a sticking probability which is proportional to k for slow
neutrons, which can be justified from the side of the dispersiontheoretic
discussion in Section 5. As energy increases, the sticking probability
goes over to 1 gradually, and we may choose the constant K to give a
smooth extrapolation from low to highenergy values, from (83a) to
(83b) . The resulting neutron cross section behaves like
c n = icX 2 P ,i{k/K)~l/v
at the lowest energies, where only the s wave, with I = 0, contributes
and gradually falls with increasing energy until it approaches the geo
metric cross section.
It is to be remembered that all this refers to the cross section for actual
formation of a compound nucleus. The "potential scattering" term
e id sin 5 of (68) is present in addition. Part of the compound nucleus
100 A Survey of Nuclear Reactions [Pt. VI
formation cross section may indeed lead to reemission of a neutron
with just the incident energy, a process of elastic scattering after com
pound nucleus formation. 1 This part can in fact give coherent inter
fering contributions to the diffraction effects. Such interference terms
can complicate the shadow scattering. In principle we would expect
angular distributions of elastically scattered neutrons to show a more or
less isotropic part, coming from the reemission of absorbed neutrons
from several overlapping levels in the compound nucleus, together with
a more complex and usually wellcollimated part, which would contain
the diffracted shadow waves plus interference terms from both sources.
In the statistical theory, however, we expect the part of the cross section
of compound formation we discuss to approach the value wR 2 for high
energies, though the total cross section, including the elastic diffraction
or shadow scattering, will become very nearly twice that.
(b) CJiarged Particle Reactions. Here the Coulomb barrier introduces
an added complication. We begin with the last member of (80) :
<n(e) = 7rXi 2 2(2Z + 1)P„{« (80)
It is now in the spirit of our statistical approach to give the sticking
probability the same form for charged particles, once they have reached
the nuclear surface, as we found it to have for neutrons, for the nuclear
forces are predominant there. If now we write the W.K.B. expression
for the penetrability P v j from Eq. (37), we have, in the region E/B « 1,
where the W.K.B. method is reliable:
<r,(c) = ttX 2 2(2Z + l)fc, • (^) V 2c < = ,rX 2 2(2Z + l) e " 2ci T^?~
Now the constant K can be chosen to make the cross section for the low
energy, highbarrier region, where the W.K.B. expression is adequate,
go smoothly over to the highenergy limit, which is just
aiic) = ttX 2 2(2Z + l)P n ,i » x£ 2
with the same arguments about the critical angular momentum l c as
in the case of the neutron. We finally obtain then, for incident charged
particles, i,
d{c) = xX 2 £ (21 + l)e 2Cl for all energies (86)
1=0
A long literature exists on the points we have treated here quite heuris
tically. By and large, the simplest justification for our treatment is
found in the related discussion of Section 5, but compare also the papers
1 Cf. discussion following Eq. (45).
Sec. 7B] Some Typical Nuclear Reactions 101
of Konopinski and Bethe (K8) and Bethe (B13), whose results we have
essentially set out above, but by rather different methods.
We include here a number of graphs (Figs. 12 to 14) showing the re
sults of computations of the cross section for compound nucleus forma
tion, based on (83) and (86), and using the same value both for the target
radius {R — I A A ^10 13 cm) and for the radius of the projectile.
X ^proton
Fig. 12a. Cross sections for protons, for formation of the compound nucleus, plotted
linearly.
The graphs apply to protons, alphas, and neutrons over a wide range
of energies, for typical target elements over the upper twothirds of
the periodic table. Following these graphs, we include graphs (Fig.
15) of the specific widths, the functions defined in (50b). These graphs
are based on the cross sections given in (83) and (86), and on the as
sumption concerning the statistical level density of compound nuclei,
given in (31). From the specific width graphs we may compute the
function D p = I^/STy = /p/S/y for any definite reaction, and hence
the total reaction cross section on the statistical theory. The special
problems raised by the deuteron and the gammaray will be discussed
later. The specific widths are given for typical nuclei as functions of the
energy of excitation of the compound nucleus, for the several possible
emitted particles.
102
A Survey of Nuclear Reactions
[Pt. VI
0.2
0.4 0.6 0.8
X = "proton /^barrier
1.0
Fig. 12b. Logarithmic plot of proton cross sections.
Fig. 12c. The proton barrier height needed for the crosssection graphs, as a function
of Z. The parameter x is given in terms of proton kinetic energy measured in the
laboratory frame. Several values for R have been assumed. In the crosssection
curves we use R = 1.4(A M + 1)10 13 cm.
xE a /E barrier
Fig. 13a. Cross sections for alphaparticles, for formation of the compound nucleus,
plotted linearly.
103
104
A Survey of Nuclear Reactions
[Pt. VI
a!
0.5 0.6 0.7 0.8
0.9 1.0
Fig. 13b. Logarithmic plot of alpha cross sections, for energies below the barrier.
x E^/E h!Lrrier
15
>
.SP 10
5 
10
20
30
40
50
Z
60
70
80
90
100
Fig. 13c. The alphabarrier height needed for the crosssection graphs, as a function
of Z. The parameter x is given in terms of alphaparticle kinetic energy measured
in the laboratory frame. (Lowest barrier has been used in crosssection graphs.)
10
\\ °
£ 5
b
1
— — °
o experimental points for Ag
£=90 O
Z=GQ
1 1
Z=30
i i i i i
123456789
tf„(Mev)
Fig. 14. Cross sections for neutron absorption in three elements.
105
10
106
A Survey of Nuclear Reactions
[Pt. VI
The same calculations which lead to the specific widths give (before
integration) the energy spectrum of emitted particles. We have already
shown a typical set of spectra in Fig. 6.
11
13
15
9
R (Mev)
Specific widths for neutron, proton, alphaparticle, and gammaray emis
7
E,
Fig. 15.
sion from several compound nuclei, as a function of energy of the emitted particle.
The specific widths are computed for residual nuclei of even A and Z. For oddeven
or evenodd residual nuclei, multiply the plotted width by 2; for oddodd residual
nuclei, multiply plotted width by 4. Interpolation for other Z can be made directly.
The application of these formulas to all the reaction types listed, in
the appropriate energy region, is straightforward. To point out a few of
the more general consequences, and to provide a model for calculation,
Sec. 7B] Some Typical Nuclear Reactions 107
we shall discuss one or two particular reactions which can be compared
with experiment.
(c) Competition. The reactions (a,ri) and (a,2n) on Ag 109 have been
studied by Bradt and Tendam (B20, Gl). Figure 16 is a simplified
version of their experimental results, and of some later work on similar
300
O In(oi, n)
7 Ag i? = 1.4(A 1/s + 4Vs). NT 18 cm
Ag(a,«)A ;
§200
0)
>
§100
Sn i? = 1.5^ 1/a 10 13 cm
14 15 16
E (Mev)
Fig. 16. Comparison of experimental and theoretical cross sections. The circles
refer to the measured relative cross section for In(<*,n) (see reference Gl), and the
triangles to the work of (B20) for Ag(a,n). Two theoretical curves are fitted: one
with R = 1.5A^lO 13 cm, and one with R = 1.4 (A* + 4«)10 13 cm. The
difference is a fair indication of the limit of reliability of our statistical theory as
well as of the assumption R ~ A^.
reactions. The cross section for formation of the compound nucleus,
with sticking probability near unity for these energies, is plotted as well
(the graphs of Figs. 12 to 14 were used). The compound nucleus can
emit a neutron, two neutrons, charged particles, or gammarays. The
nucleus is excited by the energy E a + B a . The binding energy of the
alpha, B a , is not known, but can be written B a = — 4B„ + 28 Mev,
where B n is the average binding energy of each of the nucleons brought
in, and the mass defect of the alpha is known to be 28 Mev. We cannot
give B n with any accuracy in this region of the table, but it is pretty
surely between, say, 4 and 8 Mev. The compound nucleus is then
108 A Survey of Nuclear Reactions [Pt. VI
excited to between 15 and 20 Mev in these experiments. Such an excita
tion energy brings us well into the region of dense levels. The escape
of a gammaray will be negligible here, for, as we shall see, the gamma
widths are always small compared to heavyparticle widths when the
excitation is appreciable.
Charged particle widths will also be small compared to neutron widths,
even though the excitation energy is a good deal beyond the barrier for
protons, because, from expression (52) the release of a proton with the
full available energy is not probable. It is probable that the emitted
particles take off energy corresponding to the nuclear temperature, here
some 2 Mev, and for such protons the barrier is still effective. We
expect that neutrons will go off almost all the time: T n ^> T p . The
results of experiment agree nicely with this idea from the lowest energies
measured up to about 15.5 Mev. Here the rate of increase of the cross
section for (a,n) drops sharply. With alphaparticle energy of 4 Mev
higher, the value of tr a , n has fallen to a third or less of the value of
<r a (c). This is the typical effect of competition. The factor D n has
changed from very near unity to something much smaller because the
threshold for a new reaction has been reached at 15.5 Mev. The new
reaction is the (a,2n) reaction, which leaves the residual excited nucleus
In 111 . We may estimate the probability of this reaction very roughly
as follows: The cross section for formation of the compound nucleus is
<r a (c), given in Fig. 13; £ is about unity; and the fractional number of
neutrons emitted with energy E n is, from (52), just
— — = E n <r n e E " /T (87)
In order to emit a second neutron, the first neutron must leave the
residual nucleus sufficiently excited to emit the second one. This will
occur whenever the residual nucleus has even slightly more energy (by
a few kev) than the binding energy B n of the neutron in the residual
nucleus, for the gammawidth is then negligible, and charged particle
widths are all extremely small because of the Coulomb barrier. We
can then write
In(En) dE n
a(a,2n) = a a (88)
" En, max
I n (E n ) dE n
where the integral in the numerator is carried from zero removed energy
to AE = E n max — r 2n , with V 2 „ the threshold energy for emission of a
second neutron. Both integrals can be approximated by using the
Sec. 7B] Some Typical Nuclear Reactions 109
Maxwell distribution of (87) :
1  (1 + AE/ T )e AE/T
<r{a,2n) = <r a (89)
1
(i + ^=).
max/T
where AE = i? max — r 2 „ is the excitation energy surplus beyond the
threshold of the 2n reaction, i? max = E a + B a — B n . The threshold
for the observed reaction in silver is 15.5 Mev, and E m&x ~ 2 or ZAE, so
21
OJ 1
£1
Bi 209
i2 = 1.4(A 1/3 + 4 1/3 )10" 13
Ni 60
■ a
m
i? = 1.3U 1 /3 + 4 1 /3).i ii^^ / ,. (a,2n)
(a,w)+(a,2w)
□ (<*.n)
10 20 30 40
E a (Mev)
Fig. 17. Comparison of theoretical contact cross sections with the sum of observed
(a,n) and (a,2n) cross sections. The solid curves are the theoretical values and are
marked with the radii assumed. Ni 60 data from (G2); Bi 209 data from (K5).
we can set the denominator equal to 1, and the fit with theory is then
excellent up to 18 or 19 Mev, as far as the experiments go. Agreement
was obtained by choosing the best value of the temperature r, which
turned out to be the very reasonable value 1.8 Mev. Our simple model
(Table 4) gives this value for silver excited about 18 Mev, which is not
far from what we would expect for the excitation of the residual nucleus
to the limit of the neutron spectrum, with alphas of about 15 Mev.
Other multiple reactions, as (n,2n), and even the more highly multiple
reactions, may be treated in a similar fashion. The literature contains
discussions of the most characteristic features of many other reaction
types (see Appendix II). We include more experimental comparisons
in Fig. 17; the entry of successive competing reactions is beautifully
shown (G2, K5).
110 A Survey of Nuclear Reactions [Pt. VI
SECTION 8. THE DEUTERON AS A PROJECTILE
We have not so far discussed what is the most commonly used nuclear
projectile excluding pile neutrons. This is the deuteron, 1H 2 . It is
essentially different from the proton and the alphaparticle, because
neither is it a simple nucleon, nor is it so tightly bound, like the alpha,
that its state of lowest internal energy alone plays a part in nuclear
reactions at ordinary energies. On the contrary, the deuteron is so
loosely bound — only (2.23)/2 Mev per nucleon — and so large a structure,
with its constituent nucleons often 3 or 4 • 10 13 cm apart, that these
properties play the major part in determining the course of deuter on
induced reactions.
The use of deuterons to produce nuclear reactions, especially for the
copious production of radio isotopes, is by now traditional. An excellent
set of data (C7) has been published which gives the practical yields —
often quite large — from deuteron bombardment at moderate energy.
We give a few typical numbers in Table 7. Perhaps the most evident
TABLE 7
elected Experimental Yields with
14Mev Detjt
(ThickTabget)
Yield
Product
(nuclei/10 6
Reaction
Isotope
deuterons)
Mg 24 (d;a)
nNa 22
500
Na 23 (d;p)
nNa 24
1400
AF(d;p,«)
u Na 24
5.7
p 3I W;p)
15P 32
860
Cr 52 (d;2n)
2S Mn S!
140
Cu 63 (d;p)
29CU 64
270
Br 81 (d;p)
36 Br 82
220
Te 130 (d;2n)
63l 130
95
Te 130 (d;n)
63 I 131
180
consequence of deuteron bombardment is the high excitation of the
compound nucleus formed by deuteron capture when it takes place.
The nucleus is excited by the kinetic energy of the beam plus the full
binding energy of a proton and a neutron, reduced only by the weak
2Mev binding of the free deuteron. This amounts in general to some
14 Mev plus the kinetic energy. Such an excitation implies the validity
of the statistical theory. The penetrability may be calculated as in
(37) and in Figs. 12 to 14 in a fair approximation to the deuteron contact
cross section, for a deuteron of energy Ed on a nucleus of charge Z is
Sec. 8A] The Deuteron as a Projectile 111
that for an alphaparticle with energy (2^E d ), but with a target nucleus
of charge Z/2 Vi , i.e., an alpha with E' = \AE d , Z' = 0.6Z. (This
implies the rather rough approximation, A proportional to Z.)
Such deuteroninduced reactions would lead nearly always to neutron
emission, and often to multipleparticle emission. The neutron can
take off sizable energies because of the high nuclear temperatures. It
is nevertheless striking that many reactions have been observed, espe
cially with the heavier nuclei, in which proton emission was of com
parable frequency with neutron emission. This is the case not only at
higher energies, where the proton barrier might be unimportant for the
nuclear temperature involved, but even at moderate energies, and espe
cially in the heaviest nuclei. We should like to discuss this phenomenon
in detail; it is the result of breakup of the deuteron, and it is called
stripping.
A. The OppenheimerPhillips Reaction : LowEnergy Stripping
The deuteron moves toward the nucleus in the presence of the large
electrostatic field produced by the nuclear Coulomb charge. The proton
in the deuteron is repelled from the nuclear surface, but the neutron feels
no such force. For the highest energies, or for light nuclei, when the
Coulomb barrier is unimportant, the deuteron is polarized by the Cou
lomb field, but it may not break up. In cases where the Coulomb barrier
is important — say comparable to the binding energy / of the deuteron —
the proton will not reach the nuclear surface when the neutron does.
The Coulomb forces will repel the proton, and the nuclear forces will
seize the neutron. The compound nucleus then will be formed not by
the capture of a deuteron, but only by that of a neutron, and the proton,
which will never have come within the range of the nuclear forces at
all, will fly off with a gain in kinetic energy, both from the recoil of the
broken deuteron bond, and from the Coulomb repulsion. The ratio
of the barrier height to the deuteron binding energy will be a very rough
measure of the importance of the process. This process is sometimes
called, after the first authors to recognize it, the OppenheimerPhillips
process (01).
The process can be treated by dividing it as usual into several steps.
We write the cross section for a reaction initiated by a deuteron of
energy E d resulting in the release of a proton of energy E p in the interval
dE p as follows :
a(E d ,E p ) dE p = <r d (po,R)lT(E p ) dE p (90)
Here a d is the contact cross section for penetration of the proton com
ponent of the deuteron, not to the nuclear surface, but to the more or
112 A Survey of Nuclear Reactions [Pt. VI
less welldefined distance p to which the proton has the maximum
probability to penetrate; T(E P ), called the transfer factor, takes into
account the gain in energy of the proton as it moves out, and £ is the
sticking probability of the neuteron averaged over all available levels
into which it may be captured.
The modified penetrability function for the deuteron, ad(p R), taking
into account the polarization produced by the Coulomb field, has been
given, with useful graphs (VI). The factor £ will increase with energy
of excitation, perhaps as rapidly as the level density, and will become
unity for high excitation energy, which means low proton energy. High
proton energy will mean low excitation energy, and may indicate the
existence of individual levels. For low deuteron energies, the excitation
of the nucleus will be quite low, even less than that following slow
neutron capture, since the binding energy of the deuteron will be lost to
the compound nucleus. The neutron sticking probability can be quite
low at such "negative kinetic energy" but, even then, the easy escape
of the proton from its rather distant point of closest approach will result
in a large OppenheimerPhillips cross section and a small deuteron
capture. This cross section will be larger than deuteron capture just
for this reason, even when the neutron does not always stick.
For higher deuteron energies, the proton can approach the nuclear
surface rather closely, and the most probable distance, p , will not be
much greater than R. Then the ordinary deuteron capture and neutron
emission will have an excitation function much like that of the Oppen
heimerPhillips (d,p) process. But, by reason of the high excitation
following capture of a deuteron, the (d,p) process dominates the (d,n)
process up to energies well above the barrier, since the (d,pri) process
is very likely to occur after deuteron capture.
B. Stripping Reactions at Higher Energy
Even at very high energies, of course, the passing deuteron is dis
torted by the nuclear Coulomb field, and the OppenheimerPhillips
process takes place. Here one may think not so much of the purely
electrostatic forces as of the passing deuteron in a varying field due to
the nucleus. In a frame of reference in which the deuteron is at rest,
and the heavy nucleus is moving almost undeflected past it, the electric
field experienced by the deuteron is a highly transverse field, owing to
the Lorentz contraction of the nuclear force lines. This pulse of electric
field can be Fourieranalyzed into a collection of nearly transverse
waves of a wide spread of frequencies. These waves behave nearly
like quanta, and are often called "virtual quanta" in this method of
calculation (Dl). The virtual quanta may induce photodisintegration
Sec. 8B] The Deuteron as a Projectile 113
of the deuteron as they pass by. In the laboratory system, where the
deuteron is moving, the process becomes an electromagnetically induced
stripping. It is in general a minor contributor to deuteron reactions
compared to specifically nuclear effects at high energy.
But, much more important in the highenergy limit, where the deu
teron may be handled classically, with a welldefined trajectory, is a
process we may call collision stripping (S7). Here the deuteron is
broken up by actual contact of one of its particles with the nuclear
matter, while the other particle flies off with the momentum it had at
the instant of collision, the resultant of its share of the deuteron center
ofmass motion and of the internal motion within the deuteron. This
process yields neutrons and protons in equal numbers, and in a cone
restricted to the forward direction, with a halfangle given by the simple
relation 6 ~ (W d /E d ) V2 , and energies given by E p ~ E n ~\(E d V2
± Wd Vi ) 2 , where W d is the deuteron binding energy. The process is
evidently one which leads to different energy and angle distribution from
the lowenergy OppenheimerPhillips reaction, or from emission after
formation of a compound nucleus. Although detailed agreement with
this simplified form of the theory is to be expected — and has been veri
fied — only for really highenergy deuterons, at 190 Mev, the process
must contribute continuously at lower energy, and finally merge with
the standard OppenheimerPhillips case, in which the "stripping"
occurs with by no means negligible reaction between the captured
nucleon and the one that goes free, and in which the trajectory is spread
heavily by diffraction (P2).
With deuterons in the region of moderate energy, say from 5 to 20
Mev, the stripping process is still dominant. Forwardpeaked and even
more complicated angular distributions of neutrons from these projec
tiles on a variety of targets of nearly any range of A indicate that neither
compound nucleus formation nor a statistically treated stripping, in
which classical ideas are used and all orbital angular momenta are
regarded as contributing, can completely account for the process (B23).
With good energy definition of the incident beam, it is possible to fix
upon a group of outgoing protons of a definite energy. These protons
arise from the capture of the neutron into one specific level of the prod
uct nucleus. At these moderate energies the excitation of the nucleus
may be less even than that following capture of a free neutron, because
the deuteron binding energy must be supplied to free the outgoing pro
ton. This implies that individual levels will separately contribute.
Since the incident deuteron momentum in the stripping approximation
point of view is just equal to the sum of the momentum of outgoing
proton and of the neutron before capture, forwardemitted protons may
114 A Survey of Nuclear Reactions [Pt. VI
carry most of the available momentum. The internal deuteron motion
does not often permit large values of momentum for both nucleons.
Thus the captured neutrons frequently have long wavelengths, and
quantum effects become decisive. The loose deuteron structure means,
moreover, that large values of the orbital angular momentum of neu
tron with respect to nucleus can be important, while only a few I values
can be captured into a single level. Then penetration effects may show
up strongly in the angular distribution of the stripped protons, espe
cially at small angles. Study of the fine structure of these distributions
determines the partial waves captured, and becomes a powerful means
of level spectroscopy (B22).
The really complicated behavior of the deuteron as a projectile, which
arises from its loose structure and internal motion, seems qualitatively
explained by the various processes outlined; it is by no means clear in
full detail.
SECTION 9. RADIATIVE PROCESSES IN NUCLEAR
REACTIONS
The allimportant role which radiation plays in the deexcitation of
atomic states is diminished in nuclear reactions by the numerous alter
native means of decay. In nearly all reactions, nevertheless, gamma
rays are observable products, since particle emission cannot always be
expected to leave the nucleus in the ground state, and, once the cooling
nucleus has dropped below the energy content representing the threshold
for particle emission, only radiation can take away the remaining energy.
With the new machine sources of continuous gammaspectra, moreover,
nuclear reactions induced by gammarays are of high interest. The
fact that the electromagnetic interaction between field and charge
current is completely known makes the study of nuclear structure
through radiation, rather than particle collision, seem attractive. The
somewhat illusory nature of this argument comes both from the ex
perimental difficulty of accurate measurement for processes of generally
low yield under conditions of poor energy resolution and from the com
plex character of the chargecurrent vector within nuclei, which turns
out to depend on rather fine detail of nucleon motion, as on the presence
of exchange forces and other phenomena connected with the intranuclear
motion of the meson cloud. But the whole subject is full of interest.
A. The Multipole Classification
Just as the emitted particles are classified by the orbital angular
momentum of their partial waves, which determines angular correla
Sec. 9A] Radiative Processes in Nuclear Reactions
115
tions and penetrabilities, so it is useful to classify radiation from any
chargecurrent system contained in a limited volume of radius R in a
similar way, according to the successive terms of a general expansion.
This expansion is called the multipole representation, and it amounts
to a sorting by angular momentum and parity. The electromagnetic
field is a vector field, with some special properties due to the zero mass
of the particles of the field — photons — and it turns out that there are
two independent partial waves (analogous to the familiar two types of
polarization) for each value of total angular momentum radiated away.
Speaking physically, one would expect three possible angular momentum
values for each orbital value, since the photon has unit spin. But the
condition of transversality, which is closely related to the zero mass
of the photon, excludes one orientation. In Table 8 are presented the
TABLE 8
Multipole for Given Transition
Parity
Radiated Angular Momentum, J y
Change
1
2
3
L (even)
L (odd)
Yes
No
None
None
Elec. dipole
Mag. dipole
Mag. quad
ripole
Elec. quad
ripole
Elec. octo
pole
Mag. octo
pole
Mag.
2 L pole
Elec.
2^016
Elec.
2 i pole
Mag.
2^016
selection rules and the type and order of multipole for a given parity
change and radiated angular momentum. The radiated angular mo
mentum is of course to be taken in the usual sense of the vector model :
if Ji and J/ are initial and final angular momenta, the relation J t  + J y
> Jf >  Ji — J y  must be fulfilled.
In general only the lowest order of multipole allowed by the selection
rules contributes; e.g., if the transition is Ji = 2, odd —►«// = 1, even,
we expect electric dipole only. The magnetic multipole of order L is
reduced in intensity with respect to the electric dipole of the same order
in the ratio (h/McR) 2 . For electric and magnetic dipole this is easy to
see by simply writing the ratio of the familiar dipole moments:
elec. dipole moment eR
mag. dipole moment eh/Mc
116 A Survey of Nuclear Reactions [Pt. VI
It holds more generally. The absence of any J y = forbids zerozero
transitions completely, with one quantum emitted. For a transition
in which a magnetic multipole fulfills the selection rules with the lowest
value of multipole order L, the next higher electric multipole may con
tribute radiation less by an order of magnitude, since the reduction
from order to order in L is estimated to be in the ratio (R/\) 2 <
(p/Mc) 2 ~ (h/McR) 2 . If the lowest contributing multipole is of elec
tric type, the magnetic multipoles are negligible.
1. Reduction of the Nuclear Dipole Moment. Classically, and even
in atoms, the electric dipole transition is the most intense for radiation
not too small in wavelength compared to the dimensions of the system.
From our selection rules we could expect to see forbidden transitions,
slow and nondipole in character whenever large angular momenta have
to be radiated. This is of course the origin of the wellknown nuclear
isomers, and of certain forbidden lines in the nebular atomic spectra.
But there is a general argument which indicates that electric dipole
transitions will not be so important, at least for nuclear gammarays
up to some 10 Mev.
The interaction, energy between a system of charges and the electro
magnetic field may be written :
#int = 2] Aji = X) A(pVi)
i i
In the usual dipole approximation, we replace the operator v* by pi/m,
and use the familiar relation between matrix operators, p = m di/dt.
Then the dipole moment operator becomes
Se t pi
D = SejTi cc
mi
But plainly it is only the net displacement of the charges with respect
to the center of mass which produces radiation from an isolated system.
Then the operator whose matrix element between initial and final states
determines the rate of radiation is
P= X. — (X.X) with AMX = MX> p + MX)x„
all particles TYli p n
where M is the nucleon mass, for a nucleus with A nucleons, Z protons,
and therefore N = A — Z neutrons. Here x p is the vector position of
the pth proton, x„ that of the nth neutron, and X is the coordinate
Sec. 9A] Radiative Processes in Nuclear Reactions 117
vector of the center of mass, all in an arbitrary reference system. Now
we can write
1
M
! ( i D? x " e f? x "
MA„ MA n
(91)
This is as though we calculated the dipole moment considering each
nucleon with actual charge e* to have an effective charge only e, — Ze/A.
Clearly, if all the particles of the nucleus had e t /m t  = e/M, there would
be no electric dipole radiation whatever; the charge center and mass
center would coincide, and the mass center clearly could not oscillate.
In less extreme cases, where the motion of protons and neutrons is very
similar — if they stick, say, more or less tightly together in alphaparticle
subunits — the electric dipole transitions would be at least much dimin
ished. There is no such general restriction upon the other multipoles.
Indeed, magnetic multipoles will contain contributions both from the
intrinsic magnetic moments of the nucleons and from other magnetiza
tions arising out of the transient currents coming from meson flow
within the nucleus. These can be shown to be an inescapable conse
quence of exchange forces of various kinds (SI). In the deuteron
photoeffect these currents are probably observable at low energy, where
the major contribution is from a magnetic dipole transition. In general,
then, we expect the magnetic multipoles to give somewhat larger con
tributions than the estimate made earlier would indicate, and we look
for the electric dipole term to be much reduced, perhaps so much that
the electric quadripole and magnetic dipole will overshadow its effects.
2. The Sum Rules. This reduction of the dipole moment is limited
by a very general result, an extension of the ThomasReicheKuhn sum
rule long used in the atomic case. We can write for the integrated dipole
absorption cross section
r x 2^ 2 e 2 h ,
I <T a (Ey)dE y = — — — 2^ f on
J Mc „
Now, independently of how the socalled oscillator strengths, /„„ =
 X on fK n  2 , vary with the energy difference between ground and ex
cited state, E n — E = h 2 /2MX „ 2 , the value of the sum is just
M mmj/ei eA 2
2, Jon = 2, — T~ \ J Wltn M any particle mass m;
„ Sm, i,j 2e \nii nij/
(92)
provided only that the system absorbing the radiation consists of par
ticles of charge e,, mass m,, and that the forces between them are ordi
118 A Survey of Nuclear Reactions [Pt. VI
nary forces. Exchange forces contribute a correction of similar size.
Applying (92) to the nuclear case yields the result
/•" 27rVh 1 / _ e 2 \ 7T 2 / h \ 2 , NZ
I <r d (E y ) dE y = ( 2 23 — o I = ( ) m p<?
J c A¥ p V t£2e 2 / 137 \M p cJ P A
NZ
= 0.058 Mevbarn (93)
A
From (92) it follows that a nucleus with N = Z, consisting of a set of
infinitely wellbound alphaparticles, would have a strictly vanishing /
sum. This is the result the previous section predicted. But, since
actually we take the nucleus to contain not fundamental alphaparticle
building blocks, but protons and neutrons, perhaps bound with finite
forces into alphaparticlelike units, the integrated cross, section will be
given by the form of (93). Taking the two results together, we can
reconcile them by observing that the cross section will remain low, and
we shall have little or no contribution to the / sum, as long as the energy
of the gammaray is insufficient to break up any strong correlations into
alphalike structures. But the dipole transitions cannot really be
prevented by any such internal binding of finite strength; they can
merely be deferred. Sooner or later, as energy increases, transitions are
made to states lying high enough so that any given subunit is broken
up, and the / sum begins to grow, reaching finally the total given by
(93), which any system of protons and neutrons must eventually show,
whatever their internal motion. (We exclude exchange forces for the
moment; they change nothing qualitatively.)
From these general considerations we would expect that the gamma
ray transitions from nuclei would be electric dipole, magnetic dipole, or
electric quadripole in most cases, with the last two having possibly
somewhat higher probability, for transitions involving only a few Mev
(Wl). But, for higherenergy gammaabsorption or emission, say
from 15 to 20 Mev — energies large enough to excite any transient con
figuration of nucleons, even the stable alphaparticle — the dipole transi
tions begin to show their deferred dominance and lead to integrated
dipole absorption cross sections of the order of an Mevbarn (D5), for
nuclei of middle A. This indeed seems to be a fair picture of what is
still a murky experimental domain.
Further extension of the sum rule type of calculation leads to rough
information on the values of quantities like I a y {E y )E y n dE y ; these,
taken with the experimental data (HI, K3), tend to confirm the picture
of the last paragraphs at least roughly (L4).
Sec. 9B] Radiative Processes in Nuclear Reactions 119
B. Calculation of Radiation Widths
The familiar formula for the rate of radiation of a quantummechanical
system with dipole moment D radiating light of frequency v is
Energy /second = —  D»y 
o c
Introducing a selfevident notation for the type of multipole involved,
and suppressing numerical factors near unity because of the roughness
of our estimates, we may write the resulting width for electric dipole
(2 x pole) radiation:
Bjit now we can estimate that, for a single particle moving in a region of
radius R with charge e, the dipole moment is Z);/ = eR. For electric
multipoles of successively higher order we can similarly estimate that,
apart from numerical factors, the radiation intensity is reduced for each
successive order in the ratio (R/X) 2 . Magnetic multipole moments are
smaller than the electric moment of a given order by the factor (h/McR),
which, using the nuclear radius value R = 1.5A^10 13 cm, leads to
21 x
[l35(2Z + l)] 2
(94)
1 /RV
ec r^ f I
— 137 \X/
p mag ^ J , ] ji elec r^ Q Q2^ — Mp.eleo
— \McR/ ~
These estimates will be reasonable ones for the lowlying states of
nuclei if the independentparticle, Hartreelike model is not too mis
leading. (We have suppressed factors depending on I, which would
arise in a more consistent calculation from the complex angular behavior
of the higher multipole radiation.) It seems not inappropriate to apply
these formulas for the rough computation of those longlived lowlying
states of nuclei which are responsible for isomerism. The detailed
comparison with experiment is beyond the scope of the present work;
we make no correction for the additional nonradiative transitions due
to internal conversion, and so on. But it is useful simply to show that
the radiation widths we compute would give the possibility of lifetimes
like those observed for reasonable values of radiated angular momentum
120 A Survey of Nuclear Reactions [Pt. VI
and energy. Table 9 shows lifetimes like those observed for the very
transitions which the shell model predicts for the lowest states.
TABLE 9
Lifetimes for Radiative Decay of LowLying States
A = 100 A = 200
Multipole Energy = 200 kev Energy = 100 kev
Electric 2 3 pole . 029 sec 0.9 sec
Magnetic 2 4 pole 1 . 5 years 21 years
We cannot expect the estimates of (94) to hold in the region where the
level density is very high, and the idea of the excitation of a single
particle no longer plausible. The sum rules themselves show that the
estimate of a constant dipole moment, independent of energy, must fail ;
the / sum would not even converge. A very rough idea of what to expect,
consistent at least with the notion of a compound nucleus, may be gotten
in this way. We expect the singleparticle estimate to apply not to one
single level of a highly excited nucleus, but to a whole group, dividing
up the width, so to speak, among a great many levels each of which
shares some part of the combining possibilities with the ground state.
But, over how big an energy range must we spread the radiative width?
A guess is afforded by the spacing of levels near the ground state, where
it is not unreasonable to think that only a single particle has been
excited. That spacing measures the energy region to be assigned to the
excitation of one particle. Purely as a very rough orienting estimate,
let us write for the radiative width of an average level in the region
where levels are dense :
1 /R\ 21
r elec = _\ E x
137 \X/ 7
p(0)
Ti™* = 0.02A~ 2A Ti
elec
P (E y )[l3(2l+l)] 2
(95)
where p(0) and p(E y ) are of course the number of levels per unit energy
(really of levels with certain fixed J values) at excitation energies
and E y .
The actual radiation from any level formed in a nuclear reaction
will of course involve transitions not simply to a fixed state below, but
to all combining levels at lower energy. The total radiation width then
for a given type of multipole radiation can be written
r t ot' ,e ' ec = Z Tf^iEi  Ey)
all/
Sec. 9C] Radiative Processes in Nuclear Reactions
121
Replacing the sum by an integral, and using the form of (95), we get
l,e
P(0)
P\E y max)
E _2l+l
P(E.
y max
x
Ey) ClEy
(R/hc) 21
137[l3(2Z+ l)] 2
(96)
The integrand represents the gammaray spectrum immediately follow
ing decay, without taking into account any of the subsequent cascade
gammarays. The most common example of such radiation is that
following thermal neutron capture. The spectra as measured (K7)
show very marked effects of transitions to a few specific lowlying levels,
such as the ground state itself, and do not fit our statistical estimates
very well. There is not much information about the wide gammaray
spectra which do seem to underlie these special lines and which presum
ably correspond to what we have estimated (D2).
We tabulate, for nuclei in various regions of mass number, the com
puted radiation width following neutron capture, for radiation of various
multipole types. The level densities are taken from (31), and the whole
calculation is very rough. There seems here, too, to be evidence against
TABLE 10
Total Radiation Width after Neutron Capture: Various Multipoles
A
Max
Ey = Bff
(Mev)
Widths in Electron Volts
T\ elec
p mag
r 2 el
J^mag
1 observed
60
120
200
8
7
6
4
0.4
0.1
5 X 103
0.3 X 10 3
0.07 X 103
2 X lO 3
0.2 X lO" 3
0.05 X 10" 6
30 X 10 " 7
2 X 10 7
0.3 X 10" 7
0.2
the full contribution of electric dipole radiation. Probably there are
such transitions; some appear to have been identified in light nuclei,
but they do not exceed in probability either magnetic dipole or electric
quadripole by as much as an order of magnitude.
C. PhotoInduced Reactions
From the theory so far given, the behavior of gammarayinduced
nuclear reactions follows in a somewhat sketchy way. The general
122 A Survey of Nuclear Reactions [Pt. VI
statistical theory describing the reaction as a twostep process is here
applied :
a y , p = S y (E)D p (97)
where the cross section S y is the absorption cross section for the gamma
ray, and the factor D p , as in Eq. (40), describes the breakup, by emission
of particle p, of the compound nucleus — here just an excited state of the
target nucleus — exactly as in particle reactions. Below the binding en
ergy of a single neutron, no particle reaction can be observed, of course,
since D p is exactly zero. Elastic and inelastic scattering of gammarays
are possible, and would be governed by an analogue of the usual atomic
dispersion theory, with the electric dipole moments perhaps not pre
dominant. As the particle threshold is crossed, D p rising from zero, the
cross section <r yp rises as well, probably remaining rather small though
increasing with a fairly high power of the energy, as a result of the im
portance of electric quadripole transitions. Only at energies where the
excitation of the alpha subunits becomes likely will the dipole moment
assert itself; there the cross section S y rises to high values. As soon as a
few Mev of excitation are available beyond the threshold for two
neutron, or even neutronplusproton, emission, this process will effec
tively eliminate the simple initial (7,^) Thus a peak will be observed
in the excitation function, with a width governed largely by competition.
This peak, coming wholly from the factor D p , is superimposed on a
fairly rapid rise and subsequent fall of the dipole cross section S y . There
is still no clear division of the observed peak into the two factors; it is
now fairly sure that the simple competition cannot be the entire reason
for the width of a few Mev of the socalled "resonance" for the (y,n)
reaction on a variety of targets (K4). The growth in neutron yield for
a given gammaenergy, examined as a function of A, does, however,
appear to be due mainly to the decreasing neutron binding energy and,
hence, increased opportunity for twoneutron emission, as A grows.
That the compound nuclear state formed by gammaexcitation might
be of a rather special kind, with a particular internal motion, has been
suggested (Cll, S10, T2). No clear evidence for this view has been
produced which could not be duplicated on the present picture. The
overall { <r(E) dE, the presence of an apparent resonance due to com
petition, and the variation of neutron yield with A cannot distinguish
the special motion from rather generalized features of any dipole ab
sorption. Good measurements of the peak widths and the relative
yields of the various reactions are not yet at hand.
That the assumption (97) is not entirely right seems demonstrated by
the rather high yields of charged particles, especially of protons, from
Sec. 10A] Nuclear Fission 123
gammarayexcited nuclei. These yields are far in excess of those
calculated from statistical theory whenever, as for medium or heavy
nuclei, the statistical emission of protons is much reduced by the Cou
lomb barrier. The protons tend to be distributed in angle more trans
versely than isotropically. All of this suggests that a process which
contributes only a small part of the total reaction — for medium nuclei,
the protons are only a few percent at most — can occur in which protons
are as frequent as or more frequent than neutrons. Any process which
amounts to the leakage of the protons out of a small region of the nucleus
before that region has lost its excitation energy by conduction of the
"heat" to the whole nuclear volume will do. A direct photoeffect, in
which the single proton is ejected from the smooth nuclear potential
well, is an extreme case of this view (B15). Something between this
and the statistical equilibrium idea is more likely to be correct. Emission
of alphas, deuterons, and other fragments seems to give further signs
of the need for a more detailed theory.
Reactions may be induced by the varying electromagnetic field of a
charged particle. This problem has been discussed in terms of the
method of virtual quanta, mentioned in Section 8, mainly for electrons.
SECTION 10. NUCLEAR FISSION
No nuclear reaction type has been so much discussed, and none has
attracted so many workers, 1 as the curious reaction called fission. The
fact that this reaction involved so profound a rearrangement of nuclear
matter that neutrons were emitted in greater numbers than one per
divided nucleus permitted the chain reaction, and thus the largescale
release of nuclear energy, even under terrestrial conditions of pressure
and temperature. We shall discuss fission rather sketchily and semi
quantitatively here, with major attention not to detailed experimental
results, but to the features of most general interest. All information
here presented, without exception, is from the published literature,
much of it from the spate of studies of fission in the first two or three
years after its discovery (T6). This section does not pretend to be a
full guide to the present state of knowledge of fission.
A. The Energetics of Fission
The fission reaction cannot conveniently be written as we have written
all others: T(z,p)R. The projectile may indeed be a proton or neutron,
or the reaction may be observed to occur spontaneously, but the product
nucleus is not one of the light nuclear particles, with a heavy residual
1 See, for example, the semiannual reports of the U. S. Atomic Energy Commission.
124
A Survey of Nuclear Reactions
[Pt. VI
nucleus. On the contrary, the products of fission are nuclei of sizable
charge and mass, covering a range of scores of charge and mass numbers.
In Fig. 18 we present the yield curve for several typical fission reactions
(G3, Nl). The target nucleus has divided into two major fragments
7
U 235 + K
^ 6
—
u
0)
J \Bi 209 +d
1 lu 236 +w
(3
S 4
JTh^l+a
>
"So
* 3
a
e
& 2
O
/
2
£ 1
7
1
ll
\ 1 \
50
100
Mass number, .A
150
Fig. 18. Percentage of occurrence of product nuclear fragment of mass number A,
plotted against mass number, in three different fission reactions. The neutron fission
goes with slow neutrons; the alphainduced, with alphas of 38 Mev; the deuteron
fission, with 200Mev deuterons. Note the trend toward a symmetrical distribution,
and the invariably large spread in fragment masses.
(with a few light particles sometimes also in a kind of spray). It is clear
that the description of the emission of a particle p from an essentially
unaffected residual nucleus R is inappropriate.
But the phenomenon can be pictured in a very simple and convincing
way from the most fundamental ideas of the nuclear model, treated in a
nearly classical fashion. 1 We have throughout referred to the nuclear
matter as a kind of "quantum liquid" of nearly constant density, some
1 Bohr and Wheeler (B18) gave the first extended treatment of the theory; we
follow them rather closely.
Sec. 10A] Nuclear Fission 125
1.45 X 10 14 g/cm 3 . A heavy nucleus is then a small nearly spherical
drop of this fluid, with radius R = roA^ cm. In such a nucleus, look
ing apart from the entire complexity of quantum effects, oddeven
regularities, etc., we can recognize a static equilibrium for the con
figuration. The attractive forces which cause the drop to cohere are,
just as in a drop of water, shortrange forces for any given nucleon satu
rated by the interaction with its nearest neighbors among the nucleons.
In the roughest way, then, the binding energy of a nucleus of mass A,
charge Z is just proportional to the number of nucleons present, each
forming its full number of bonds with other nucleons. The negative of
the binding energy, — E B , we define as usual as the total energy of the
neutral atom (nucleus plus its surrounding electrons) minus the total
energy of its constituent neutrons and protons when removed to large
separations, including the energy of the electron needed to neutralize
each proton. Then stability means positive Eb. Then Eb ~ volume,
and, assuming constant density, the volume energy alone gives —Eb
= —avA. But, like any liquid drop, the nucleus has a surface. Even
for the heaviest nuclei, many nucleons lie on the surface of the drop.
Their bonds are not all saturated; there is a net deficiency in binding
energy, a positive surface energy exactly like the surface energy of a
water droplet, but clearly of the greatest importance. We then expect
— E B ~ —a (volume) + b (surface), under the same constant density
assumption. There is yet another classically evident contribution to the
nuclear energy. This is the mutual Coulomb repulsion of all the protons
in the nucleus. It is not a saturating force, with finite bond numbers
and a short range. On the contrary, it is a longrange force to which
all the protons contribute. On the most naive assumption of con
stant charge density, the Coulomb energy of a spherical drop of radius
R = r A ^ is simply the integral
/Ze\ /Ze\ rrdVdV _3Z 2 e 2 3 ZV
Vol/ ' W/ JJ I r — r' 1 ~5~R~~ 5r A* ^
sphere
Purely classically, then, the binding energy of a liquid drop of constant
density and uniform charge density 1 is given by the form
Z 2
E B = a v A + a s A % + a c — ^ (99)
These simple ideas lead to a classical theory of fission. Suppose that
now we have such a charged classical drop. If we slowly deform it by
1 For a careful discussion of this whole procedure, and of the limitations of these
ideas, see (F2).
126 A Survey of Nuclear Reactions [Pt. VI
elongating it, we clearly increase the surface area: this costs us surface
energy. But the two halves of the drop are now found at a larger dis
tance from one another than in the spherical form. This means that
their mutual Coulomb repulsion has decreased, and we have gained
binding. As we deform the drop more, we increase surface energy and
decrease Coulomb repulsion still more. Finally we can divide the drop
into two remote spheres, each of half the original charge. In this state
the Coulomb energy has been much reduced, and the surface increased.
For a sufficiently high charge density, however, the Coulomb reduction
must outweigh the surface energy increase, and the fissioned drops will
be the stable configuration. The original sphere may then be in fact
in a state of unstable equilibrium; any finite distortion from the sphere
will cause the droplet to divide. Or it may be that, originally, small
distortions into an ellipsoid will cost more surface energy than they gain
in Coulomb energy, and the original sphere will be in a state of stable
equilibrium. Yet finite displacements — distortion into something
nearer a dumbbell shape — may so reduce the Coulomb energy that some
intermediate distorted shape is a state of unstable equilibrium, capable
of going either back to the original sphere, or on to two divided droplets.
In this case the original drop is only relatively stable, and a genuine
disturbance might cause fission. We shall see that this is the actual
nuclear case at hand.
1. The SemiEmpirical Theory. We can make all these notions more
precise. Let us go back to the dropmodel energy content appropriate
for the constantdensity nucleus. We wish to evaluate the constants
for volume, surface, and Coulomb energy. In addition, we shall add
two terms which represent an effort to make the drop idea conform more
closely to nuclear experience. It is fundamental in nuclei of course that
the exact numbers of individual nucleons have a real role, unlike the
molecules in any largescale drop, where small fluctuations in mass
have an unimportant effect. There are two reasons for the effect, and
both of them reflect the essentially quantum character of the nucleus:
first, the total number of particles is small, so that quite generally small
changes in A could have important effect; and, second, the strong inter
actions seem to result in something like the atomic shells showing as fine
detail on the general course of the liquiddrop energy content. We
shall throughout disregard the details of this phenomenon. There is
a second consequence of the special nuclear forces : the protons and the
neutron seem to tend to group in pairs or even in alphaparticlelike
units. There is a binding energy bonus for having the neutrons and
protons equal in number. We shall introduce semiempirically, then,
two terms into our binding energy formula:
Sec. 10A] Nuclear Fission 127
(1) A term, called the symmetry energy, which depends on the square
of the difference between the number of protons Z and the number of
neutrons N = A — Z. The fact that the term is quadratic represents
the fit in first approximation to the trough of the valley in a binding
energy surface, plotted against A and Z. The most stable isotopes lie
in a crooked valley furrowing this surface; we imagine that the bottom
of this valley can be approximated in cross section by a parabola. If
we associate with each unpaired particle a definite energy, the whole
term can be written
(N  Zf a r
(TV "t 6) A.
(2) A term to stand for the last unpaired particle, expressing the fact
that nuclei of even N and even Z are the most stable for a given A, and
that those with N odd, Z odd are the least stable, while the oddeven
or evenodd category is intermediate. This term may be empirically
determined from the difference in energy between successive beta
transformations down a chain. With these additions, the semiempirical
formula becomes
z * W  zf
E B = ~a r A + a s A V3 + a c ~ T .+ a r ± 8(A) (101a)
A^ A
We can evaluate ay and as by fitting the formula to the known mass
defect curve, which is by no means very accurate in the region above
A = 40 or so. The procedure is made easier if we first evaluate a T in
terms of a known a c . This we can do by fitting the valley of the stable
isotopes, i.e., by finding the Z which corresponds to the most stable isobar
for each A. The data fix this for odd Z at least to within about ±J^
for Z, making the valley pass between the two most abundant isobars.
From our formula we need find only dM(A, Z)/dZ and set it equal to
zero. This should mark the trough of the valley. [Note that M(A, Z)
= E B + AM n + Z(M H  M n ).] From dM/dZ = we get a relation
between Z and A. The function so determined, called Z A , is given by
the expression
Z A _ (0.00081 + a T )
A ~ 2a T + 0.00125A %
using ac = 0.584 Mev, as given below. If we smooth over the data,
neglecting several kinks in the Z A function, we can find that, with ac
as given, if a T = 0.083 mass unit, the fit to the empirical course of the
curve is good. We consider throughout only nuclei above A ~ 20. A
128 A Survey of Nuclear Reactions [Pt. VI
set of good values for all the constants then is
„ 77.3 I A \ 2
E B (Z,A) = 14.0A + 13.1A* + 7\z ~ z )
A
A% lo;Aodd
+ 0.584 — ^ + \ ±6V A ' Z even, A even (ioib)
where we have expressed Eb in Mev.
A similar semiempirical formula due to Fermi has been extensively
tabulated (M7) for nuclei of all plausible Z and A, going far beyond
the known range. The form there used does not display the Coulomb
energy directly, but combines it with the symmetry energy, using as a
reference the bottom of the actual valley of stability Z A rather than the
artificial situation of equal neutron and proton number. Written for
the atomic mass in mass units, the Fermi form is
M(A, Z) = 1.01464A + 0.014A^  0.041905Z A
+ 0.041905(Z  Z A ) 2 /Z A
^ ±0.036M«; ! ° dd ' V Ven nm ,
+ Z even, A even (101c)
0; A odd
with Z A /A = 1/(1.9807 + 0.01496A % ). An earlier version of the
formula is cited in a more accessible place (Sll). Our (101b) is in
reasonably good agreement with the Fermi formula, but the latter
(101c) is preferable for actual calculations. (See Section 3B in Part IV
of Volume I.)
A more elaborate treatment has been given which includes a measure
of the compressibility of nuclear matter, i.e., a departure from the con
stant density assumption (F2) . The effect is quite small. The Coulomb
constant ac is not in fact evaluated from the semiempirical formula for
a large number of nuclei, but is given from the comparison of the so
called "mirror nuclei," isobars with N — Z = ±1, where direct meas
urement of betadecay energy has been made (Fl, F3). The radius it
corresponds to is somewhat larger than that we have taken from high
energy nuclear reactions; it gives r = 1.48 and not 1.4 X 10~ 13 cm.
This difference may reflect in part the tendency for the protons to move
to the outside of the nucleus, or it may simply reflect the inadequacy of
the rather crude semiempirical formula.
Let us now consider the energetics of fission with formula (101b). A
very simple model of fission may be made by assuming that the drop
divides into two drops, each of the same charge density, proportional to
Z/A. Let one fragment have charge fZ, the other (1 — f)Z, and simi
Sec. 10A] Nuclear Fission 129
larly with the A' a. Then the energy released on such a fission, say
AE(f), is given by the expression
AE(f) = 13.1A^[1 /«  (1 /)«]
Z 2
+ 0.584 — [1  f A  (1  /)»*] in Mev (102)
The volume and symmetry energies do not change; we neglect the small
term 5 of (101), since we have not required integer values of Z or A.
The treatment is entirely classical, and depends only on the assumed
surface and Coulomb energies of drops of constant charge density. We
plot in Fig. 19 the net energy release AE(f) as a function of / for nuclear
drops of several charges. Note that, above a certain value of Z, fission
will release energy for a wider and wider range of fragment sizes. For
high enough Z, indeed, very small fragments — resembling alphapar
ticles — may be released with a net gain in stability. This simple theory
predicts the occurrence of fission and of alpharadioactivity for high Z.
Below the limiting Z value of course not fission, but combination of light
nuclei, will represent a gain in stability.
So far we have discussed only the energy difference between initial
spherical drop and final spherical fragments. If this energy difference,
AE(f), is positive, the process will proceed spontaneously. But the
rate of fission, which will determine whether or not the process is ob
servable (rates corresponding to halflives of more than Ty 2 ~ io 2122
years are not now observable), is still entirely open. In Fig. 20 we have
represented quite schematically this complicated process. We have
plotted the energy difference between initial spherical drop and dis
torted drop against a single parameter — some measure of the effective
distance of charge separation. Of course the distortion can in fact be
represented only by many parameters. In our simplified plot, the value
AE is measured from the initial energy to the final energy, from E t to E f .
We have plotted four cases: reference to the figure will make them clear.
In case (1) fission is not an exothermic process and cannot occur. In
cases 2, 3, and 4, it is exothermic and occurs spontaneously. In all of
these, the initial sphere is in equilibrium at sp under the mutual action
of surface and Coulomb forces. But in case 4 the equilibrium is un
stable; any finite disturbance will cause fission. The actual nucleus
would last only a time comparable with the characteristic nuclear times,
for the zeropoint oscillations would cause the needed displacement.
Cases 2 and 3 show initially stable nuclei; small displacements will not
lead to fission, but will be followed by a return to the initial spherical
equilibrium. Only finite displacement, adequate to supply energy greater
130
A Survey of Nuclear Reactions
[Pt. VI
=M(Z, A)
M(Z,fA)
0.2 0.4 0.6 0.8 1.0
Fractional charge of fragment, /
1.2
Fig. 19a. Potential energy classically available for fission into two fragments of the
original charge density, but charges fZ and (lf)Z, plotted as a function of fragment
size, /, for several nuclei. For heavy nuclei, division into fragments of a wide range
of sizes is exothermic; for nuclei lighter than, say, bromine, fission is endothermic
even with the most favorable fragment size, and symmetric with / = 0.5.
Sec. 10A]
Nuclear Fission
131
than the height of the barrier, at P, will lead to fission. Classically, it
will never occur spontaneously; the initial state is a; state of relative
stable equilibrium. Quantummechanically, in the actual nuclear case,
spontaneous fission can occur, by the tunneling effect — diffraction
beneath the barrier. Whether or not this will occur at an observable
rate will depend on both the height
and width of the barrier. In case 2,
the barrier is high and wide; we
may expect that spontaneous fission
will not occur observably, and that
large energies of excitation compar
able with AE would need to be
added to such a drop before fission
could occur. This might be the case
of a nucleus well beyond the limiting
Z near 35, but not one of the natural
radioactive series.
It is evident that the most useful
application of these ideas would be
the calculation of the barrier heights,
or fission threshold energies. The
potential energy contour of the drop
plotted as a function of some repre
sentative parameters determining its
distortion from the sphere (of course
two parameters are all we can plot)
is a surface with a flat portion far
away from the center (where the two
fragments are far apart), rising to a
fairly high elevation as the frag
ments approach and feel the Cou
lomb repulsion; in the very center
of the contour map, where the drop is almost spherical, there is a
craterlike bowl in the center of which lies the original sphere before
fission. Small displacements will never cross the lip of the crater, but
there are passes in the crater wall, the lowest of which lies above the
crater center by just the fission threshold energy. This pass, or saddle
point, is what we want to find, in respect to both its height and its
position on the map, i.e., to the shape of the drop at the unstable equi
librium position, where it can either go on to fission or return to the
original sphere. (See Fig. 21.)
, 20 40 60 80 100
Atomic number, Z
Fig. 19b. Maximum potential energy
classically available on fission into two
drops of equal charge and mass, plotted
as a function of atomic number Z.
132
A Survey of Nuclear Reactions
[Pt. VI
50 100
Distance apart of nuclei centers (10" 13 cm)
Fig. 20. Potential energy of nuclear drop as a function of distortion. The compli
cated distortion is here represented by a single parameter, some measure of effective
distance of charge separation. The energy surface is plotted in the plane which
represents the most favorable distortion, through the "pass" in the energy crater.
Note the stable position at sp, the original spherical shape, the increase in potential
energy to reach the pass at P, the energy threshold for fission, Eth, and the long
decline down the curve to D and beyond. The distant part of the curves is simply
the 1/R behavior of the Coulomb energy of spherical charges. If continued in until
the fragments touched, any distortion being neglected, this would reach the points
marked C, the end of the Coulomb barrier. How the drop distortion allows fission
at much lower thresholds is seen from the bending over before point C.
Sec. 10A]
Nuclear Fission
133
134
A Survey of Nuclear Reactions
[Pt. VI
The full calculation is of course difficult. The number of degrees of
freedom of a classical charged drop under axially symmetric distortions
is very great, but some important consequences can be obtained quickly.
(1) Let us consider the limiting case, where the charge density is
small and the surface energy far outweighs the Coulomb repulsion. In
this case, the drop will try to minimize its surface at all deformations,
and the spherical shape will be stable for every distortion up to the very
radical one of allowing the drop to be molded into two spheres and mov
Undistorted
a; = 0.74
(a)
(b)
x = 0.65
x«l
(c)
(d)
Fig. 22. A series of equilibrium shapes for the nuclear drop. Shape (a) is of course
the original sphere; (b) the shape of the drop at the critical distortion for
x = 0.74 = (Z 2 /A)/(Z 2 /A)n m ; (c) critical shape for x = 0.65; (d) the shape corre
sponding to a: « 1, division into two nearly tangent spheres, where the Coulomb
tidal distortion is neglected (low Z).
ing those spheres just out of the range of the surface forces. The shape
of Fig. 22d is the case we shall compute. Here the Coulomb energy is not
strictly zero, but only very small. The critical shape will then still
allow for a tiny neck connecting the two fragment drops, which can be
torn apart by the Coulomb repulsion. If we neglect the energy of the
neck, which is very small in our case, we can simply compute the differ
ence between the energies of the two equal spherical fragments and the
original sphere. The fission threshold energy, E t h, will in this case be
E th = 24.Tr
[©"*]
M(A) 1A r ] 2 <r + 2
3 (Ze/2) 2
+
5(A/2)^
(Ze/2) 2 3 (Ze) 2
2(A/2y A r 5A A r
(103)
Sec. 10A] Nuclear Fission 135
(with <r = nuclear surface energy/cm 2 ) from which
E th Z 2 e 2 /r
= 0.260  0.065 f (104)
MA H r ) 2 <r ' ' A iirr 2 a
e 2 /r Z 2 _ 5 Coulomb energy
where the parameter — • — is just  • .
4irr a A 3 Surface energy
(2) Now we consider the other limiting case, with the original droplet
so packed with charge that the slightest displacement from the spherical
shape will produce fission. Here Eth = 0. To compute this we con
sider small constantvolume distortions, axially symmetric, without mo
tion of the center of mass, but otherwise arbitrary. We may represent
the radius of the drop surface as a function of the colatitude angle, 0, by
a series in the orthonormal set of Legendre polynomials
y(6)
= 1 + 0P x (cos 9) + a 2 P 2 (cos 0) + a 3 Ps(cos 0) \ (105)
R
Here the coefficient of Pi (cos 8), oti, is set equal to zero, since for small
displacements (where the higher harmonics may be expected to grow
small), it corresponds to displacement of the center of mass. One con
dition is set on the motion by this requirement. The purely mathe
matical requirement that the function y(6) be singlevalued for repre
sentation as a Legendre polynomial expansion does restrict the physical
motion somewhat. Such a shape as that of the other limiting case —
two nearspheres joined by a thin thread — cannot be so represented.
This turns out to be an unimportant restriction; our answer is very far
from such a shape for the observed fissioning nuclei.
If we make the calculation of the classical area and Coulomb energy
with assumption (105), and keep only the leading term, in a 2 , we obtain
these results for surface and Coulomb energies of an infinitesimally dis
torted drop (81), neglecting all terms of higher order in the small co
efficients a n :
5A H r \5 /
tfsurf  E surf sphere = 4x4 Hr„ a ,
to . 
(106)
isurf  ^surf •" " = 4irA "ro
Then the fission threshold energy, measured in units of the undistorted
surface energy, becomes
+ ■■■) (107)
±irA % l0 2 c 5 V # surf sphere
136
A Survey of Nuclear Reactions
[Pt. VI
We can read off the result: any displacement will lead to fission (i.e.,
Eth < 0) if JV phere /#surf Sphere > 2. Writing this result in terms of the
semiempirical constants, we have
E c /E sm!  lim = Z 2 /A \ iim a c /a s = 2 .. Z 2 /A  lim = 45.0 (108)
0.30r
0.20 —
0.10
o.e l.o
x =E c /2E a =CZ 2 /A)/(zVA) lim
Fig. 23. The fission threshold energy, Eth, as a function of the parameter x. The
dimensionless function /(x) is just Eth measured in units of the nuclear surface energy
in its undistorted spherical form, and x is defined by the relations x = Ec°/2E S °
= Z 2 /A/(Z 2 /A)um From fitting the experimental data for U 239 , we get
(Z 2 /A) v ,
■ 48 (see text). The function f*(x) is the energy difference between the
sphere and the shape of Fig. 22d, two equal nearly tangent fragments, again meas
ured in units of the surface energy. For small x, below the vertical stroke on the
curve of f(x), the function f(x) is taken from its limiting value for small Z; above
the righthand vertical mark, f(x) has been calculated numerically. Between the
two strokes f(x) has been simply interpolated freehand.
We can of course continue the series still further, and examine how
Eth/E aur f spheie , which we will write as f(x), behaves as a function of
x = Ec/2E S . This has been done out to quite high powers of the small
Sec. 10A] Nuclear Fission 137
coefficients — with as many as six or eight harmonic terms. A numerical
calculation, done with highspeed calculators (Fll), has been carried
out with ten harmonic terms, and without making any powerseries
expansion in the coefficients. The function fix) is known near x =
from the calculation in case 1 above, and near the limit x = 1 we have
plotted the calculated results in Fig. 23. The dashed part of the curve
is a reasonable interpolation. We can regard x either from a theoretical
point of view, as the ratio Ec/2E S , or in a form more directly comparable
with experiment, as (Z 2 /A)/(Z 2 /A) ]inl . In the latter form, we see that
we can read off the value of the fission threshold E t h for any nucleus by
computing the surface energy asA 2/3 , noting that E t h = f(x) ■ as A y *.
We can now apply these results best not by taking the limiting
{Z 2 /A\ Ta entirely from our semiempirical constants, but by fitting
one observed fission threshold (S8) to the fix) curve. If we fit E t h (U 239 )
= 6.67, and take, for the surface energy for U 239 , 502 Mev, we obtain
f[x(U 239 )] = 0.0132. This yields Z 2 /A\ Um = 47.8, which can be com
pared to 45.0, obtained by taking the constants directly [Eq. (108)].
2. Quantum Effects. We have treated the problem wholly classically.
It should be observed that this neglects two quantummechanical effects.
First, fission may take place for excitation energies below the fission
threshold by reason of the tunneling effect; if the representative point
comes near the pass over the crater lip, it will often go through the thin
barrier remaining. In addition, the vibration of the drop in the dis
torted mode will have a zeropoint energy. Both of these effects can be
estimated by computing how the barrier looks as the representative
point moves in its path over the energy surface. This can be done under
the assumption that the drop distorts through the sequence of shapes
near the saddlepoints for larger values of x. With this assumption,
Frenkel and Metropolis (Fll) calculated the barrier as a function of
distortion. The Gamow penetration could then be estimated by using
this potential barrier, and an effective mass estimated from hydro
dynamic arguments. They obtained a penetration probability P for
various E th given by P = io 7  85 ^. The zeropoint energy can be
calculated from the shape of the crater near zero distortion. This gives
a result of some 0.4 Mev for the uranium case and leads to an estimate
of the oscillation frequency. Such a zeropoint energy, small compared
to the excitation involved, implies that the motion can be represented
classically as a rather welldefined trajectory by building up wave
packets from the actual quantum states. The threshold estimate is
then made as follows :
"th (Classical) = £/ z eropoint l * neutron ~T~ ''neutron binding
138 A Survey of Nuclear Reactions [Pt. VI
Using the results that neutron binding in U 239 is 5.2 Mev (from the
semiempirical formulas) and that the measured neutron kinetic energy
is 1.0 Mev, we get the 6.6 Mev used above for E t h (classical).
Only qualitative results can be expected from this theory. The
actual fission is not into symmetric fragments at these excitations,
and the fission thresholds do not in detail vary from nucleus to nucleus
in the smooth way here predicted. Moreover, the spontaneous fission
rates are in poor agreement. It must be observed that the very large
exponents which occur mean that a small error in the barrier will lead
to enormous errors in the rate calculations. Perhaps only the logarithm
of the decay rate should be expected to be of some significance, within
a small factor of the truth. Arguing the other way is more sure. It
turns out from the f(x) curve (it will be an interesting exercise for the
reader) that for Z > 98 the typical isotope would be radioactive to such
an extent that it would probably disappear from the earth in geologic
time, not because of the accidents of alphaparticle disintegration rates
(which vary both up and down in just a few Z or A units), but because
of the inescapable effects of fission. Thus the fact that the heaviest
nucleus occurring in nature is uranium is a consequence only of the
kinks in the binding energy curve, coming from the closed shells at
Z — 82, N = 126; this means that the next few elements have short lives
against alphadecay. But the more fundamental reason that no ele
ments of Z = 100 or up are found is certainly fission. It will be inter
esting to see the first nuclear species whose lifetime is controlled by
spontaneous fission.
There is one more deduction from this simple theory which is of inter
est. The calculation of Eq. (104), which we can plot as the function
f*(x) for < x < 1 (see Fig. 23), gives the energy difference between
the original sphere and the nuclear matter arranged as two equal frag
ment spheres just touching. Now the energy released on symmetrical
fission is a given function of Z and A, independent, of course, of the
path followed in dividing. If the released energy is Ai?(3^), then the
energy difference between the critical distorted shape and the separated
fragments is just f(x)asA ^, while that between separated fragments and
two tangent fragment spheres is f' t (x)asA 2A , which is, from Fig. 23, some
25 or 35 Mev greater in the region of the fissionable nuclei. This means
that, when the distorted surface finally tears to release the two frag
ments, which when far away will eventually come to their stable spherical
shape, the difference of about 30 Mev is stored as energy of distortion —
energy of excitation — in the two fragments. It is, moreover, plain that
the reverse of fission — union — will in general require much more kinetic
Sec. 10B] Nuclear Fission 139
energy than that released in fission. It will be very unlikely that the
special distortion which leads over the low pass in the crater lip will be
achieved in the random collisions of nuclei.
B. The Products of Fission
The most obvious consequences of the simple theory given above are
in fact not observed. The fission fragments do not appear as equal
nuclei with / = }/%. On the contrary, the distributions of Fig. 18 are
those observed experimentally. No clear explanation of this fact has
yet been given. It seems almost sure that there is no complex distortion
type which leads to a lowlying pass for some asymmetric distortion of
the classical drop. It is much more probable that again we have to do
with a reflection of the nuclear shell structure, permitting some motions,
some ratios of charge to mass in division, more easily than others. The
closedshell N's at 50 and 82 do seem to mark nearmaximum yields.
There is some evidence that isotopes of those stable elements with
highest N — Z are favored, but the expected predominance of even
products is not yet found in the admittedly complex chains of fission
fragments. It should be obvious by now that the initial products of
fission are not in general stable nuclei, but are very neutronrich, and
begin chains of betadecays to reach the appropriate ratio of Z/A for
their smaller A (B18, W4). It has been shown that, in this betadecay
chain, excitations are in some cases in excess of the neutron binding
energies for the product nucleus of a given betadecay. This leads to
the emission of delayed neutrons whose time of emission after fission is
determined by the preceding betadecay processes. Much more numer
ous are the socalled prompt neutrons, which emerge after a time short
compared to any possible betadecay. It seems very likely that these
neutrons are evaporated from the highly excited fragments as they fly
apart after the fission act. This would imply that some two or three
neutrons are emitted, and that their energy spectrum and angular dis
tribution would be the quasiMaxwellian one of our statistical theory
(compare Fig. 6). The fission fragments, however, continue in motion
at their high speeds of near 1 Mev/nucleon for a distance of a few milli
grams per square centimeter, or some 10~ 3 cm, in solid matter. The
evaporation will occur with neutron widths measured in many kev at
least, or in times short compared to the fragment time of flight. So
the neutrons will evaporate isotropically from the moving fragment,
and in the laboratory system will be emitted preferentially forward,
since their kinetic energy will be quite comparable to that of the frag
ment, expressed as Mev per nucleon. Some evidence has been obtained
to confirm this picture as well (W9).
140 A Survey of Nuclear Reactions [Pt. VI
The products of fission, then, are mainly the two large fragments and
the prompt neutrons. The fragment decay leads to betaparticle chains,
gammarays, and the delayed neutrons. But there are other charged
particles which emerge at least occasionally upon fission. In about one
fission in five hundred an alphaparticle forms the third fragment, with
kinetic energy from a few to about 25 Mev (M2). It seems well estab
lished that these smaller fragments actually originate from the com
pound nucleus, and from a tendency to come off at right angles to the
direction of the main fragments, and that they arise during the act of
fission.
The systematic change in shape of the fission fragment sizedistribu
tion curve with energy of the bombarding particle is the most striking
feature of Fig. 18. Apparently, as the excitation exceeds the fission
threshold more and more, the tendency toward the symmetrical fission
expected in the simple theory becomes stronger. Slowneutron uranium
fission is never symmetrical ; at energies around 100 Mev, the symmetri
cal division is the most probable. Direct measurement of the ionization
energies of the fragments confirms the observation on the masses of the
fragments (J4). Whatever special effects produce the asymmetry are
increasingly unimportant as more energy becomes available to the
vibrating drop. It does not seem true, however, that the fragments of
an asymmetric division have total fission product energies higher than
the fragments of nearly symmetric division; there is even some tendency
toward the reverse.
C. Fission Cross Sections
Fission, as a mode of decay of a compound nucleus, should be produced
by any means that excited the nucleus above the required threshold
energy. This has been verified for slow and fast neutrons, for protons,
deuterons, and for alpha and gammarays. In general the cross section
can be computed by our usual methods — simply writing the cross section
for formation of the excited compound nucleus by a particle of energy
E v , and following it by the factor for its competitive disintegration:
i{W)YAW)
*,(E P ) = ,(E P ) " ' " > (109)
*/ T An "T J 7 T ' ' '
where 1/ is the width for fission from a compound nucleus of excitation
W. The calculation of the fission width is the calculation of how the
excitation of the nuclear drop is distributed among all its possible modes
of motion: there is available a large amount of phase space for motions
which lead to distortions from which the drop cannot divide; only a
small volume in phase is assigned to such motions of the point repre
Sec. 11] Nuclear Reactions at High Energy 141
senting all the parameters of distortion as do lead over the pass in the
wall of the energy crater. Below the threshold for fission, the excitation
energy can be lost by any other energetically available means, usually
gammaradiation or neutron evaporation, but not by fission. As the
excitation W = E p + Eb(p) increases to near the classical fission thresh
old, tunnel effect can lead to fission. The fission width will increase,
and fission will become a competitive reaction. We can expect the
fission width to rise rapidly, perhaps something like exponentially, as
the fission threshold is considerably exceeded. What the cross sections
will be then will depend largely on how high the fission threshold is
relative to the excitation energy of a given mode of excitation. Thus
<rf(E n ) for U 238 + n 1 is zero until a threshold of about 1 Mev, from
which it rises slowly to a value comparable with irR 2 . It stays nearly
constant because for some time 1/ and r„ rise about in the same way.
We know from the experiments with very fast fission, on elements like
bismuth, where the fission threshold is some 12 Mev or more, that the
neutron specific width rises more rapidly than that for fission. Very
high excitation in Bi 209 leads mainly to multiple neutron evaporation;
some ten or twelve neutrons are boiled off before fission becomes a very
important competitor. Only when the fission threshold has come down,
because of the gradually increasing value of Z 2 /A in this neutron boiling
off, does a/ begin to grow to a value comparable with unity, actually
about 0.1 or 0.2. This is confirmed both by the fissionproduct dis
tribution (which is that of fission by a nucleus of A ~ 200) and by the
fact that the fission fragments come apart with energy near that ex
pected from fission near the threshold energy. The distribution of
fragment sizes becomes broader with increasing excitation; the mode of
division does not need to be so special. Even so, it is plausible that there
are many more ways for a highly excited drop to emit neutrons than to
divide by fission, so that, while the I> rises rapidly, it does not rise so
rapidly as the other terms of total r, almost all of which, at these ener
gies, arise from neutron evaporation.
Special processes, like photofission and fission in the region of applica
tion of the dispersion formulas, have been discussed in the literature
(B18, G3, Nl).
SECTION 11. NUCLEAR REACTIONS AT HIGH ENERGY
The processes we have discussed so far are characterized by the fact
that the energy of the incoming nucleon, both potential and kinetic,
becomes quickly shared among all the particles of the nucleus. After
a relatively very long time, during which numerous nucleon collisions
142 A Survey of Nuclear Reactions [Pt. VI
occur, this energy may again become concentrated in a particular
nucleon, or in some special mode of motion, and the nucleus will de
excite by particle emission, fission, or the slow processes of radiation.
As the energy brought in grows, the number of available modes of
disintegration naturally increases, and the lifetime of the compound
nucleus becomes shorter and shorter. With energies in the range up to
some 30 to 50 Mev, the nucleus will emit typically several nucleons
before it cools, and the reactions are increasingly complex. Such reac
tions as (p,3n) and (p,4n) have been studied (K5). But, as the energy
increases, there is a gradual transition to a state of affairs which is best
described in a conceptual framework quite the opposite from that
suited to the compound nucleus. This highenergy point of view works
best above some hundreds of Mev of incoming nucleon energy, and the
two schemes will merge in the broad transition region between.
Most characteristic of the highenergy region is the fact that the mean
free path, in nuclear matter, of the fast nucleon is comparable with the
radius of the nucleus itself. The nuclear matter is no longer a black,
impenetrable obstacle, which traps the incoming particle, but a kind of
sphere of dilute gas, through which the fastmoving nucleon has an
appreciable chance to penetrate without any collision whatever. Even
a collision or two do not by any means rob the incoming nucleon of most
of its energy. The momentum transfer in such a collision, which is
typically transverse to the classically pretty welldefined trajectory of
the shortwavelength incident nucleon, is of the order Ap ~ h/R, where
R is the range of nuclear forces (S7). This implies an energy transfer
of some 30 Mev. In the short time of such a collision (~10 22 sec)
the struck nucleon will make only very few, if any, collisions with the
neighboring nucleons of the nucleus, and, although the momentum trans
fer is not so large that we are wholly safe in regarding the struck nucleons
as free, still that will be a reasonable initial approximation. We shall
see that the principal effects of the nucleon environment are conse
quences of the Fermi degeneracy of the nucleon matter. The collision
will be modified by the impossibility of leaving the struck nucleon in a
momentum state already occupied by some other nucleon of the nucleus.
The geometry of the approach of the nucleon will determine the se
quence of events. If it passes through the nuclear edge, it may make
no collision, or perhaps one. A very fast nucleon, having lost only 30
Mev, will emerge. The nucleus left behind may be excited by the full
amount of the energy given to the struck nucleon, or by very much less
if the initial path of the struck nucleon allowed it as well to leave the
nearby nuclear surface without further collisions. A fast particle striking
Jhe very center of the nucleus, where the nucleus is thickest, may have
Sec. 11 A] Nuclear Reactions at High Energy 143
to travel a few mean paths in nuclear matter, and will typically make
several collisions, leaving in its wake a few nucleons of 20 to 30 Mev
energy each, perhaps losing all its energy, being finally captured. The
secondary nucleons in turn can escape only if they are close to the
nuclear surface and moving outwards. Some may do so; others will
collide with many nuclear particles, and gradually spread their energy
around among many modes of motion, forming a heated compound
nucleus, which will cool off just as in the reactions of lower energy. A
typical manyparticle nuclear reaction, often called, from its appearance
in nuclear emulsion photographs or in the cloud chamber, a "star," can
then be thought of as a kind of approach to thermal equilibrium, be
ginning with a welldefined singleparticle trajectory, followed through
a cascade of secondary, tertiary, etc., collisions, and ultimately resulting
in a diffusion of excitation energy like heat through a conducting sphere.
The initial stages form the highenergy limit; here the reaction is to
be described by a stepbystep following out of the nuclear cascade,
taking into account collisions with one or with a few correlated nucleons;
finally, the energy is shared by the collective motion, in something like
transient thermal equilibrium, which is the lowenergy, compound
nucleus picture. A single star event may exhibit both features; it is the
fact that the first stage takes a time ~10 22 sec, very much less than
the 10~ 16 sec of the last stage, which permits a more or less sharp division
into the two stages. On this sharp division, which is only approximate,
the simplicity of the picture depends. The intermediate time, during
which the energy is distributed among too many degrees of freedom
to be described in detail, but too few to approach any sort of quasi
equilibrium, is beyond the present theory; such complex cases will be
especially important in the lighter nuclei, and for a mass A nucleus at
energies 10A Mev, enough to dissociate the whole nucleus. Our treat
ment will deal wholly with the two wellseparated stages: the initial
cascade and the final thermal equilibrium.
A. The Nuclear Cascade
The initial cascade has been followed out by the most direct means.
The theory has been the semiempirical one of the socalled Monte
Carlo method (G5), in which after the establishment of the model and
the appropriate cross sections— which are the decisive features for the
accuracy of the results— the calculation is done by considering the
successive events in the motion of the incoming nucleon and all its
collision partners, with their collisions in turn; choosing the actual steps
by a series of random choices; and finally cutting off the whole process
when some arbitrary lowenergy limit is reached beyond which the energy
144
A Survey of Nuclear Reactions
[Pt. VI
is assigned as equilibrium excitation of the compound nucleus. This
stepbystep process is extremely tedious, and only a few hundred indi
vidual "stars" have been followed through on paper. The results are
subject then to a fair degree of statistical uncertainty over and beyond
shortcomings of the model. But enough has been done to make the
general features of the highenergy stage in star formation fairly clear.
1. Fermi Gas Model of the Initial Nucleus. The short collision time
and high recoil energy make most plausible the use of the simplest of
V
J
Fig. 24. Diagram of the potential well in which the nucleons are bound in the sim
ple Fermigas model. The Coulomb barrier affects only the protons. The well
depth is determined by the nuclear density and the assigned binding energy.
nuclear models, that of the Fermi gas of noninteracting nucleons con
tained in a well of assigned diameter and depth. Naturally the correla
tions actually arising from the nuclear forces, here ignored, will be
important eventually. One clear sign of this is found in the process of
the pickup of a nucleon from the nucleus, to emerge bound to another
outgoing nucleon in an emitted deuteron, a process far more important
than the Fermi gas model would predict. But, aside from such details,
it is appropriate to begin with a Fermi gas.
The procedure is the familiar statistical one for finding the distribution
of a completely degenerate Fermi gas, at T = 0°. The nucleus is re
placed by a spherical potential well in which the particles move without
interacting. Each particle has a wave function which is simply a plane
wave; the finite depth of the well, and hence the leaking of the plane
waves into the classically forbidden region outside the well, is neglected.
Then a particle of each spin is placed in every momentum state until
all particles are accounted for. The neutrons and the protons are re
garded as entirely independent and as moving in separate wells; the
additional Coulomb barrier for the protons may be regarded as beginning
at the well edge. Figure 24 shows the model schematically. Replacing
Sec. 11 A]
Nuclear Reactions at High Energy
145
the sum over the filled states by an integral, and remembering that
(2s + 1) particles with different spin orientations may be placed in each
cell in phase space, of volume (2xh) 3 , we obtain for the number of
particles of a given kind, with spin s, in a sphere of volume V,
N s
(2s + l)p 2 dp cK2/(2xh) 3
where p F is the limiting momentum to which the states are occupied,
and the energy corresponding to pf, the Fermi energy E F , is given by
E F = p F 2 /2M a . If we take the sphere of nuclear matter to be a sphere
of constant density independent of the mass number A, as is usual, and
write for the radius of the nucleus R = r A H , we can easily find an
expression for the Fermi energy for the nucleons of one kind: E Ft =
(18tt) k /8 X h 2 /Mr 2 ■ (N s /A) H , where N s is either the atomic number Z
for the protons, or the neutron number N = A — Z for the neutrons.
This gives the numerical result: E Ft = 9.7(N s /A) 2A (a /r ) 2 Mev, where
we have written a /r , the ratio of a , the classical electron radius,
2.82 X 10~ 13 cm, to the nuclear radius parameter. For orientation a
few values are listed in Table 11. The well depth is now fixed to give the
TABLE 11
Proton and Neutron Fermi Energies for Various Nuclei
Ef (protons)
Ei?(neutrons)
Cu 65
n>
1.4
22.6
26
1.6
17.5
20
Cs 1:
1.4
22
27
1.6
16.9
20.7
1.4
21
28
1.6 X 10" 1S
16.1 Mev
21.5
highestlying neutron state about 8 Mev binding energy, and an addi
tional Coulomb barrier may be assigned for the protons. This implies a
nuclear potential well for neutrons in lead about 35 Mev deep, using
the smaller value of the radius parameter, and an additional Coulomb
barrier V c for the protons about 0.5(Z/A M )(a Ao) Mev high, or <~14
Mev for lead. In the calculations the two types of nucleons are often
represented by a single gas of about average properties.
In the region of high energies, the simple Fermi gas model has two
characteristic consequences which are certainly to be expected even from
more realistic models of the internal nucleon motion. It is perfectly
clear that the complete neglect of nucleon interactions is an extreme
procedure which tends to underestimate the range of momenta repre
146 A Survey of Nuclear Reactions [Pt. VI
sented in the statistical motion of the nucleons. One can regard this
lack of a highmomentum tail as implying either the neglect of spatial
correlations, in which the nuclear density fluctuates toward higher
values because of the nucleon interaction and the transient formation
of subunits of two, three, or four nucleons, or as a neglect of occasional
transitions, made possible by nucleon interaction, from the lowest
states of the Fermi gas to some of the higher unoccupied momentum
states. Both ways amount to the same thing; they form the basis for
regarding the Fermi gas calculation as a first step in a systematic approxi
mation method. We shall look here, however, into the consequences
of the raw Fermi model as they affect highenergy interaction events.
(a) Influence of the Exclusion Principle. In this model, the incoming
nucleon, once it enters the nuclear volume, fills a hitherto unoccupied
momentum state. Now it enters into a collision with some one of the
nucleons already present, lying in one of the filled levels below the
limiting Fermi momentum p F . The spirit of the model, representing
all nucleons as free, now clearly requires that the momenta of the two
collision partners after the collision lie in an unoccupied region of the
momentum space. A simple exchange of momenta corresponds to an
elastic collision. Any other event must begin by fulfilling the condition
of entering fresh momentum space regions. Evidently a small momen
tum transfer to the struck nucleon is forbidden unless the nucleon ini
tially lies very near the edge of the sphere in momentum space of radius
p F . As long as momentum transfers not very great compared to p F are
of importance in the process, which means that the incident particle
may have momentum up to many hundreds of Mev/c (or even con
siderably higher if forward scattering alone is under consideration),
the cross section for collisions with the average nucleon of the Fermi gas
will be much reduced by the requirements of the exclusion principle.
This will evidently increase the chance of passing through nuclear matter
without collisions, and will tend to reduce the relative probability of
smallangle scattering, both near zero degrees in the centerofmass
frame and, because the projectile also satisfies the exclusion principle,
near 180° as well.
Figure 25 presents the geometrical considerations in momentum space.
A calculation (G5) on this purely geometrical basis shows that the
exclusion principle reduces an assumed isotropic cross section for scatter
ing in the ratio: ,  2 \
_ / 7 PF 2 \
0"av — Cf r ee II— —  1
\ 5 pi 2 /
where o free is the isotropic scattering cross section per unit solid angle for
a free nucleon, and p t is the incident nucleon momentum. This formula
Sec. 11 A] Nuclear Reactions at High Energy 147
holds only for values of p t > 2 V2 p F ; below that incident momentum the
result is more complicated. The direct analytical evaluation has not
yet been performed for a case with the cross section anisotropic. The
angular distribution can be approached through the same geometrical
scheme, if the scattering cross section is taken as a function of the mo
mentum transfer  p {  p f  =  5  only, which corresponds to the Born
approximation without exchange.
Fig. 25. Kinematics of the collision between an incoming nucleon and the nucleons
of the Fermi gas. The incident, recoil, and final momentum vectors are shown, with
spheres of radius in momentum space just p F , the Fermi momentum. The shaded
region is excluded; no collision in which the vectors Precou or p flna i terminate within
this region is permitted by the Pauli principle.
(b) Influence of the ZeroPoint Motion. The angular distribution of
particles scattering from the Fermi gas involves two distinct and im
portant effects: the exclusion of lowmomentum transfer collisions, and
the smearing out of the otherwise unique correlation between scattering
angle and energy of emerging particle, which can be regarded as the
effect of the zeropoint motion of the struck nucleons of the Fermi gas,
which of course spread over a wide range of momenta, and hence of
relative velocities, as seen from an incoming nucleon of fixed energy.
The geometry of the event is just that of Fig. 25, and one can write
rather complicated expressions for the cross section in the laboratory
system for scattering a particle into the element of final momentum of the
scattered particle, p/  .
¥ = 'WtPF, P/, 6) (HO)
148
A Survey of Nuclear Reactions
[Pt. VI
where/ is a function of the momenta and scattering angle, and the regions
of integration for the total cross sections can be found from the figure.
Since the cross section for neutronproton scattering, for example, is
far from simple in the relevant energy regions, only numerical calcula
tions have been made. Figure 26 shows the general results to be ex
pected: the curves correspond to special choices for the cross section.

10°/ \

45°
20V''
/ V
/ \ \
\^
50
Energy (Mev)
100
Fig. 26. The energy distribution of nucleons scattered at a fixed laboratory angle
by a single collision with a nucleon in the nucleus. The effects of the Fermi motion
and of the exclusion principle have been included. The incident energy is 100 Mev;
an empirical fit to the neutronproton scattering cross section was used. The vertical
strokes mark the energy expected at the three angles shown if the collision was
with a free stationary nucleon, in which case the energy would be a unique function
of angle. The spreading effect of the Fermi motion is conspicuous (65).
The smearing of the energyangle correlation and the shifting of the
scattering toward rightangle scattering is at least made qualitatively
evident. All this applies of course to an event in which only a single
primary collision is made within the nucleus.
2. Following the Cascade: The Monte Carlo Method. The absence
of any theoretically based analytical form for the nucleonnucleon
scattering cross sections, and the rather complicated geometry in coor
dinate and momentum space required to treat the problem, have led
to the use of an interesting method which enables not only calculation
of the results of a single collision, but also actually follows out step by
step the development of what may be called a nuclear cascade.
In this application of a general computational scheme, called for
obvious reasons the Monte Carlo method, a single incident nucleon of
given energy is allowed to enter the nuclear volume. Its collisions are
Sec. 11A] Nuclear Reactions at High Energy 149
followed in complete kinematical detail, and the subsequent collisions
of all its collision partners are followed as well until either: (1) any of
the excited nucleons passes outside of the nuclear volume, after which
of course no further interest attaches to it except that it is an emitted
nucleon, with its energy, direction, and charge; or (2) any of the nucleons
involved, while still within the nuclear sphere, reaches an energy below
some previously selected value, after which it is regarded as captured,
and its excitation energy passes eventually into the motion of an excited
compound nucleus. Then follows the process of nucleon evaporation,
regarded as separate from the initial cascade. The cascade computations
are usually made graphically. The distinguishing feature of the method
is that the decision as to which of a set of equally probable events occurs
is made by some procedure of random choice, using tables of random
numbers, spinners, etc. Various simplifications of geometry may be
made, and the continuous functions of angle, etc., which occur are
typically stepwise approximated by dividing the interval into a set of
discrete subintervals, giving the function some constant value as an
approximation throughout the whole of the subinterval.
Let us follow a neutron of energy E incident on a nucleus of fixed A
and radius. Where does it make its first collision? Its point of entry
is chosen by making a random choice among the rings of equal area
normal to the direction of the beam. Then its distance of travel is
found by assuming that it moves with a mean free path given by the
assumed total interaction cross section with free nucleons of the given
density (the effect of the exclusion principle can come in only after the
collision has been made). A distance 5 which is small compared to X,
the mean free path in nuclear matter, and which gives a probability of
collision 5/X, can be chosen, and a random decision made as to whether
or not it collides in this line segment. The nucleon continues to progress
segment by segment in a straight line, until either it leaves the nuclear
volume or it makes a collision. Typically at these energies a fair fraction
of the nucleons traverse the nucleus without collision, especially those
which enter near the edge. When the nucleon has made a collision, a
whole set of random choices must be made. First, the momentum of
the struck nucleon drawn from the Fermi gas must be found, by pick
ing at random from intervals of equal probability in momentum space,
weighting the flat Fermi momentum distribution by the dependence
of the collision cross section on relative momentum. Then the angle of
scattering must be found. Usually it is best to work in the centerof
mass system, and to approximate the differential cross section for scatter
ing by a dozen or so steps. Once the angle of the collision has been found
by a random choice, the final momentum vector of each of the collision
150 A Survey of Nuclear Reactions [Pt. VI
partners is fixed uniquely by the conservation law, since the collision
is regarded as free. Now one may see if the collision is allowed by the
exclusion principle. If both the final vector momenta do not lie outside
the filled sphere, the collision is nullified, and the original particle con
tinues undisturbed until it makes another collision. If the collision is
permitted, then each of the collision partners is now a nucleon of definite
vector momentum at a specified point in the nuclear sphere. The
history of each can now be followed until it in turn escapes or is absorbed.
Account is of course kept of all partners in each collision, and the event
associated with one incident nucleon is not complete until every struck
nucleon has been emitted or absorbed. It is obvious that the com
putation is tedious; it is equally clear that it is an extremely flexible
method, capable of high accuracy. It is hardly a calculation so much as
an experiment in thought, using a welldefined model.
Two calculations have been worked out on these lines for incident
energies 90 and 400 Mev, and reported in some detail (BIO, G5). They
are unfortunately not strictly comparable, mainly because different
fundamental cross sections for collision were used. In one case the target
nucleus was lead, whereas in the other it was A = 100, more like the
heavy nuclei of photographic emulsion. In Fig. 27 is drawn a typical
cascade event just as it was followed. Here the approximation was made
of treating the nucleus as a twodimensional circle, after the first collision;
the graph then gives directly the projected angles of emission. In the
event shown a 400Mev nucleon entered, making its first collision about
onethird of the way across the nucleus. The two partners of this colli
sion were nucleons initially of 431 and 13 Mev of kinetic energy. It
will be noted that a nuclear potential well of 31 Mev is assumed, with an
additional barrier of 4 Mev (half the Coulomb barrier, since protons
and neutrons are not distinguished). One of the first pair of collision
partners leaves the nucleus directly, as a fast cascade fragment of 282
Mev. The other goes on to try to make a collision in a short distance, at
the point marked by a circle, but it is forbidden to do so by the exclusion
principle. Then it goes on to an allowed collision with a rather fast
nucleon of the Fermi gas. In all, this event consisted of six nucleon
collisions: one nucleon enters the nucleus with 400 Mev kinetic energy
outside; and three leave, taking with them over 300 Mev, mostly in the
kinetic energy of the one very fast collision partner of the first collision ;
while four more are excited but "captured," leaving the residual nucleus
excited by 66 Mev, with the mean binding energy per particle taken at
about 8 Mev.
The collection of results for both calculations gives some impression
of the nature of nucleon cascades. The mean excitation energy of the
Sec. 11A] Nuclear Reactions at High Energy 151
nucleus changes very slowly with the energy of the incident particle:
the 90Mev neutron on lead left a mean thermal excitation energy of
some 44 Mev, the 400Mev proton on emulsion nuclei only about 50
Mev. This slow rise of excitation with incident energy is partly an
artifact of the somewhat forced comparison, but it is actually to be
28 Mev
7 Mev
400 Mev
282 Mev
Fig. 27. A projected diagram of a nuclear cascade event, followed by the Monte
Carlo method. The entering nucleon has 400 Mev of kinetic energy; the numbers
mark the energy of each of the nucleons participating. The path of each nucleon
is shown until it leaves the nucleus, or until it reaches an energy below 35 Mev,
taken as the threshold for capture. Along the path of each particle an effective
collision is marked with an open circle; a cross implies a collision which was for
bidden by the exclusion principle. Note that in this event three cascade particles
emerged, all in the forward hemisphere, and the nucleus was left excited thermally
with about 65 Mev, enough to evaporate several neutrons isotropically before be
coming stable. From (B10).
expected as a general result, since the highenergy nucleons make high
energy secondaries, which typically depart from the nucleus with most
of their energy, since the mean free path rises with nucleon energy.
In both calculations also there is a rather wide distribution of energies
among the cascade products, and a considerable fraction of the energy
almost always goes to them, as the low mean excitation indicated.
With the lower energy and the bigger nucleus, however, about 5 percent
of those incident particles which made any collisions whatever left their
whole energy in nuclear excitation, with no cascade particles emitted.
In the highenergy, A = 100, case, no event of this kind occurred at all
152 A Survey of Nuclear Reactions [Pt. VI
(among sixty studied); the minimum energy found in the cascade
particles was about half the incident energy, 200 Mev.
The general nature of the cascades can be seen from Table 12 and
TABLE 12
Nuclear Cascades Studied by the Monte Carlo Method
90Mev 400Mev
Neutron Proton
on Pb on A = 100
Fraction of incident particles traversing
nucleus without any collision 0.15 . 33
Fraction of cascades without emerging
particles 0.05
Mean thermal excitation energy 43 Mev 50 Mev
Mean number of cascade particles
emerging per cascade event 1.2 3.2
Maximum number of emerging cascade
particles 3 <~68
Figs. 26 and 27. It should be remembered that the sampling error
alone in all these calculations is still quite large. Only a few hundred
cascade particles have been "seen to emerge," and information about
any special category of them is necessarily only qualitative in character.
It is quite clear from the results that this process does not much
resemble the evaporation model. The large number of emitted particles
with quite high energy, and the strong departure from anything like
isotropic distribution in angle, are distinguishing features of the nuclear
cascade. The present experimental material is fully consistent with the
model we have outlined here. Of course, the evaporation of lowenergy
particles, among them often complex nuclei of A from 2 to 10, with
isotropic directional distribution, is expected and observed to follow,
completing the deexcitation of the struck nucleus (Bll, H3).
In both calculations reported, the neutrons and protons were not
distinguished, but replaced by a gas of nucleons with about average
properties. This reflects the feeling that the cascade particles will
roughly be randomly divided into neutrons and protons, without much
correlation between particle type and either direction or energy. The
fact that in heavy nuclei, especially, there are more neutrons than pro
tons is perhaps the principal difference to be recalled, apart from the
collision cross sections. The Coulomb barrier and the binding energy
differences, especially the symmetry energy, which tends to keep the
nucleus from becoming either proton or neutronrich in excess, will be
unimportant factors in modifying the cascade, because of the typically
Sec. 11B] Nuclear Reactions at High Energy 153
high energies of the emitted nucleons, in strong contrast to the evapora
tion mode of disintegration. Some detailed examination of the Monte
Carlo results tended to verify the idea that nucleons emitted would
be about half protons and half neutrons, for the nucleus with A = 100
and a 400Mev incident proton. It is important to keep this feature
of the cascade in mind when discussing the experimental data, which
for the most part give information concerning only the ionizing prongs
of the star event. There is ample empirical evidence, however, that the
neutrons are in fact emitted (C8).
The actual collision cross sections for free collisions are still of course
not entirely certain. The empirical situation is discussed in Part IV,
Volume I, but the principal question comes from the very curious ob
served behavior of a p . p , which is so strikingly independent of both energy
and angle above about 100 Mev. Whether this property is one character
istic of the special states allowed by the exclusion principle in a proton
proton collision, or whether it reflects some deeper difference between
the behavior of protons and neutrons, is a matter which will much affect
the easy treatment of nucleons on an equal footing. It is this flatness
of the cross section which is responsible for the increase of the proton
energy loss in nuclei of medium size at high energies, and consequent
growth in star size as the nucleon energy goes from 100 to 400 Mev,
even in medium A nuclei.
B. Correlations among Nucleons
Of course, the neglect of all nucleon interaction in the Fermi gas model
is to be regarded as a zeroth approximation. In fact, the nucleons are
not uniformly distributed either in momentum space or in coordinate
space. The nuclear matter is lumpy, with constantly forming and dis
solving groups of nucleons. When a given nucleon enters into a collision,
it is not free. Some momentum may be transferred to a neighbor through
the force binding the two; or the recoil may be regarded as shared by a
correlated spatial cluster of two or even more nucleons. In the complex
process of the cascade, with high momentum transfers and repeated
events, these effects show up only as eventual improvements to our
present crude theory; but there are already known processes in which
these correlation effects are, so to speak, the whole story. In the single
elastic scattering of fast protons by light nuclei (where the cascade has
no room to develop), in processes of typically low momentum transfer
(as reactions induced by gammarays and by the absorption of slow
mesons), and in the socalled "pickup" processes (the inverse of deuteron
stripping, in which a fast nucleon leaves the nucleus no longer alone but
now paired with another nucleon which originated in the target, to
154 A Survey of Nuclear Reactions [Pt. VI
form a stable fastmoving deuteron), the influence of the correlations
and their consequent momentum distribution is decisive.
The probability for finding two nucleons of the Fermi gas at positions
r x and r 2 respectively is of course given by the integral
^(ri, r a ) = £ fdr 3  ■ dr N \ *( ri  • r N )  2
ap J
where the wave function >P is given by the determinantal expression
^
<Pb(ri)<f>b(r2) ■
,<Pa(*j) =^e*""x(o>)
Using the orthogonality of the individual plane waves for different
allowed momenta hk a , and replacing the sum over values of k a by an
y /•Pmax
integral over momentum space £ ~~ * ; I dp, we obtain the
k a (2irh) 3 J
result
P(ri,r 2 ) = 22 — 2 (Ufa)
where the o li2 give the spin, and t x , 2 the character, of the nucleon at
points t\ and r 2 . The space function w depends only on the distance
between the nucleons and is given by
,  3(sin 5 — 5 cos 5) .
w(\ ri  r 2 ) = w(8) = h5 = p max  rj  r 2 
o
(Hlb)
Thus there is no correlation in position between unlike nucleons, or like
nucleons differing in spin; but between nucleons alike in type and spin
there is a definite spatial correlation arising from the exclusion principle,
even if all forces between nucleons are neglected. Presence of interaction
forces of course affects the wave function and hence the correlations.
Using the Fermi gas as a zeroth approximation, calculations have been
made to find what the spatial correlation is like under various assumed
forces. Figure 28 shows the general character of the results. The
momentum space distribution is given with and without interactions.
The details of course depend on the forces assumed to act, but the gen
eral effect is to replace the rectangular Fermi gas momentum distribution
with one which contains some components of higher momentum —
traceable in the usual perturbation theory to transitions to unoccupied
states of the Fermi gas spectrum induced by collisions. The actual
distribution, with interactions, is somewhat as though the Fermi gas
Sec. 11B] Nuclear Reactions at High Energy
155
were present but not at zero temperature. A calculation by Watanabe,
assuming certain Gaussian potentials, indicated that the ground state
of the nucleus of high A might correspond roughly to the momentum
distribution in a Fermi gas of nucleons at a temperature of 6 or 8 Mev
(W2).
There are many processes, mentioned above, which reflect the mo
mentum distribution of the nucleons in the nucleus. One of the most
Fermi gas
Fig. 28. The momentum distribution of nucleons in a Fermi gas at zero excitation,
and the same distribution when interactions are considered. Note that the effect
of interactions in momentum space simulates the presence of thermal excitation.
interesting is the rather unexpected process called "pickup." Table 13
gives a summary of some experimental results which make it plain that
the pickup process is by no means a rare one (Y4).
TABLE 13
PickUp Cross Sections fob 90Mev Neutbons
C Cu Pb
"■inelastic, all events
0.2
0.8
1.8
barns
Relative cross section for
events yielding:
Cascade protons with
E p > 20 Mev
40%
30%
25%
Pickup deuterons
{E d > 27 Mev)
12%
7%
4%
Evidently the escape of the deuteron intact is relatively more difficult
for the larger nuclei. It is to be expected that a deuteron can leak out of
a goodsized nucleus at these energies only if it is formed in the far sur
face region; otherwise one or the other of the nucleons within it will
collide in leaving the nucleus. This implies that the formation of such
156 A Survey of Nuclear Reactions [Pt. VI
a system is really quite probable when the whole nucleus is taken into
account.
The process can be described in rather classical language as the en
counter of the incoming nucleon with another nucleon, of the right type
and moving with the right direction and speed, so that the two move
along together, forming a deuteron when the binding forces become
effective. The whole event is by no means localized in a small region of
space within the nucleus, since the deuteron is a big structure. Pre
sumably the nuclear edge contributes most of the pickup processes,
when nucleon passages outside the nuclear volume by distances as great
as the range of the nuclear forces often contribute. Energy and mo
mentum must be pretty nearly conserved between the two particles,
since a strong recoil would imply the breakup of the weak deuteron
bond. The whole picture is roughly confirmed by experiment; deuterons
do appear in fair number, coming off mostly forward, with about the
energy of the incoming particle. At other angles the effect is smaller,
and the typical energies lower. The incoming nucleon seeks a partner
moving with the same momentum; then the two could move with
negligible relative kinetic energy, and deuteron formation would be
easy. But such a process would rob the nucleus of a great deal of energy,
since essentially it would then emit two particles each with kinetic
energy equal to that brought in by only one. The overall energy and
momentum conservation then implies emission of somewhat slower
deuterons.
The process is treated by an approximation that is physically very
satisfactory. We make two assumptions: (1) The incoming neutron
interacts with only a single proton at a time. All multiple events, sub
sequent collisions, etc., are neglected, to be treated separately. We will
find the cross section for a single proton and simply multiply the result
by Z. (2) During the collision, the forces between the collision partners
are so strong that they overwhelm all the forces which bind the struck
nucleon into the nucleus. The whole effect of the binding forces, which
are by no means neglected, is thought of as determining the initial
momentum distribution of the struck nucleon in the original nuclear
ground state. For this second assumption, the approximation has been
given the name of the impulse approximation; it has found application
in a variety of problems involving collisions with a complex system (Cl).
We write the usual form for the cross section per proton from the
timedependent perturbation theory:
2x . .„
d*df = — Pf\ H of  2 dE
n
Sec. 1 IB] Nuclear Reactions at High Energy 157
Call the final deuteron momentum, measured in the centerofmass
system of the struck nucleus, hK; the initial neutron momentum hk .
Write the energy of excitation of the residual nucleus W/ — Ef — E ,
and the deuteron binding energy B. If we normalize the continuum
states of the initial neutron and the final deuteron centerofmass motion
to unit volume, we have for the cross section per proton, for deuteron
emission into solid angle dfi leaving the nucleus with excitation Wf,
2tt ,__,„. 2* M 2MhK ,
C
1 M 2 K
^=^1^0 = .— F 0/ N0
27T 2 h 4 k
■dSl\V of \ 2 (112)
The matrix element  V f  is that of the neutronproton interaction
taken between the initial and final states normalized as described.
These would be unitamplitude plane waves for the incident neutron and
the outgoing deuteron centerofmass if we used the Born approximation,
but the present calculation is not restricted to that simplification. For
present purposes, we will simplify by neglecting the change of deuteron
momentum with Wf, which is good in the limit of high neutron momen
tum, and take merely some average Wf. Now let us take as given the
undisturbed wave function for the proton in its initial state, bound to
the nucleus. Call this &(%), writing it as usual as a function of the
proton position, r p . We can find from this by Fourier transformation the
momentumspace wave function,
"^T^/^,*,
(2x)
where now  <pi  2 is the probability for an initial proton momentum k p .
The chief assumption of the impulse approximation now tells us that
this initial momentum distribution alone affects the process, which in all
other respects behaves as though the proton were initially free. Then
overall conservation requires the two conditions :
(a) K = ko + k
(113)
h 2 X 2 M 2 _
(b) = B\ + W f
W 4M 2M
We will also write the wave function for the neutronproton system in the
initial and the final states as fe, ^ respectively. Here each is a "deu
teron" wave function: in the initial state, the "deuteron" is of course
158 A Survey of Nuclear Reactions [Pt. VI
highly excited, with a large positive energy, while in the final state the
outgoing deuteron is bound, in its ground state, \f/ , of internal motion.
These functions, like the interaction potential, we take to depend only
on the relative coordinates of neutron and proton; the centerofmass
coordinates simply lead to the required momentum conservation. In
the initial state the relative internal momentum of the neutron and
proton is given by (k p — k )/2 = K/2  k . The factor of onehalf
arises in the usual way from the introduction of the reduced mass in the
equivalent onebody problem expressing the relative motion. Now
the squared matrix element of the interaction, which determines the
cross section, can be written, for a single proton,
 H of  2 =  w (kp) I 2 X  fdr+ *(r n  x p )V(x n  r p )fo(r re  r p )
(114)
The content of the impulse approximation is just the writing down of
this expression, first, to be summed over all the protons, and, second,
as a simple product of the twoparticle scattering matrix element with
the probability for a given proton initial momentum. Now we can write
the wave functions and scattering potential V(i n — r p ) in the momentum
representation, giving them as a function of the relative momentum,
say q, and the matrix element can be written
(0 V \D)
^o(q), ^(q)^
»(fkb)
(115)
Thus the scattering amplitude in this twobody collision determines the
whole process, using (114). In principle, this amplitude could be ob
tained by a complete knowledge of neutronproton scattering experi
mental values, though the most important range of q is not easily studied
in this way. The energy conservation (113) can be used to fix  q . A
related calculation has been carried out numerically by Heidmann (H5).
It is instructive to apply the Born approximation to our somewhat
more general result. For simplicity let us neglect spin questions, and
consider only the triplet states of the original collision, putting a factor
of % into the crosssection formula (112). Then we will replace the
initial neutronproton wave function fe by a plane wave, giving simply
a deltafunction in momentum space. We can write for V just T n . p — E
and the matrix element for the scattering, (0  V  D), becomes
(B + h 2 (K/2  k ) 2 /M) X «,(K/2  k )
The operator T — E has been transposed, so that it acts on the final
Sec. 11C] Nuclear Reactions at High Energy 159
state wave function <p . With this done, we can write the entire quantity
 H f  2 , which we need to sum over all the protons, in the form
D  H„  2 = Z\ w  2 X (B + h 2 (K/2  k ) 2 /M) 2 X  * (K/2  k )  2
V
(116)
The first factor Z\ </>i\ 2 is simply the number of protons to be found with
initial momentum K — k in the original nucleus; the second factor is
the energy operator; and the third factor just the probability for a
definite relative momentum q = K/2 — k in the deuteron ground state.
With a reasonable choice of \f/ , say the familiar Hultheh (C2) approxi
mation, iAo(r) = const. (e~ ar — e~^ T )/r, it turns out that the last two
factors roughly compensate, so that the variation of as with neutron en
ergy is dominated almost wholly by the proton momentum distribution.
The cross sections given by the impulse approximation certainly
account for the rather surprising frequency of pickup deuterons, and
for their distribution in energy and angle. Indeed, the calculations
apparently overestimate the frequency of pickup, presumably because
subsequent interactions break up the nascent deuterons. But it is most
satisfactory that this process, so unexpected at high energy, can, espe
cially in just that region, be given such a simple explanation in terms
of momentum transfer.
The pickup process is the first of many found to lend themselves to
the impulse treatment, and hence found to depend very much upon the
momentum distribution within a nucleus : elastic scattering of nucleons,
photon absorption, and many meson processes, like production by
nucleons and gammas and capture by nuclei, are in this category (C3,
F15). The description of the nuclear matter in momentum space, both
by statistical and by more detailed methods, is sure to become more
important and more familiar, supplementing the present description
given most frequently in coordinate space only.
C. The Optical Model for the Scattering of Nucleons
At high energies, as indeed even at lower ones, the simplest of scatter
ing experiments is the measurement of the overall attenuation of a
beam of incident nucleons. If the experiment is done under conditions
in which the energy is welldefined, and with socalled "good geometry"
(i.e., an arrangement such that particles which have deviated even
slightly from the original beam direction are excluded from detection),
it is possible to measure also the fraction of the beam which has been
elastically scattered. The convenience of such experiments has led to
the use of a quite abstract and general nuclear model, which is tailor
160 A Survey of Nuclear Reactions [Pt. VI
made to give elastic and total cross sections without at all worrying
about the details of the nuclear collisions. In an earlier section we saw
how the assumption that the interaction between neutron or proton and
nucleus was strong led to the idea of a sticky nucleus, opaque to the
passage of nucleons, and then to a total cross section which for energies
sufficiently high became just twice the geometrical cross section of the
nucleus. The elastic scattering was represented by the portion of the
emergent wave which produced the ordinary shadow behind the nuclear
obstacle by interference with the incident wave. For a black obstacle
we saw that this gave a coherent elastic, or "shadow scattering," cross
section which was just equal to the inelastic cross section %R 2 . Its
angular distribution was also determined. The optical model here used
(F6) is a straightforward generalization of the results obtained earlier,
extended to the case where the obstacle is no longer black but "gray,"
for the free path for nucleon collision in nuclear matter is no longer
negligible, as we have seen above, in the hundredMev region. The
ideas of physical optics, however, in the familiar KirchhoffFraunhofer
approximation, become better and better because %/R <3C 1.
For this model, we think of the nucleus as a gray refracting sphere.
Its opacity, or optical density, and its index of refraction for the Schrod
inger waves characterize the model. Then the amplitude of the scattered
waves, and the damping due to absorption, can be computed, generally
neglecting all interface reflections, and considering only "volume"
effects, in analogy with the W.K.B. approximation. In its simplest
form, the model regards all nuclei as uniform gray spheres, which differ
only in diameter. This geometry is of course too simple, and in no way
required by the optical approximation. It may be better to think of
the nuclear matter more realistically as thinning out gradually toward
the edge from a uniform central core; there is some evidence for such a
model in exactly this sort of calculation.
The nuclear matter is assigned a complex refractive index, N = n + it.
Here n is the ratio of the nucleon wave number k at a given point within
the nuclear volume to the wave number k Q = (2ME) 1A /h of the incident
nucleon outside in free space (neglecting Coulomb forces). The quan
tity r specifies the amplitude damping within nuclear matter; the wave
amplitude changes by a factor 1/e in a traversal of a distance within the
nuclear matter equal to one wavelength of the incident particle, \(/(x)
= ioe~ xk ° T , and is related to the mean free path for interaction in
nuclear matter I — y^k^r. The inelastic scattering which we will calcu
late includes all exchange and absorption effects; it depends only upon
the imaginary part of N, that is, on t. The real part of the index of
refraction will determine the coherent scattering, in which the scattered
Sec. 11C] Nuclear Reactions at High Energy
161
wave represents particles of the same type as those of the incident beam
and having exactly the same energy in the centerofmass system. We
will consider only the uniformsphere model; for this, N is constant
within the nuclear sphere of radius R, and unity everywhere outside.
In the shadow of the sphere, the transmitted wave which passes at a
distance from the central diameter of the sphere (see Fig. 29) emerges
with an amplitude a(p)e' koR , where e lk ° B is the amplitude of any portion
of the incident plane wave which misses the sphere. The region of the
shadow must contain a wave of lessened amplitude whose intensity
Beam
Source plane
(a)
(6)
Image
plane
Fig. 29. The geometry of the optical model for nuclear scattering. In (a) the beam
is shown incident on the nuclear sphere of radius R. The path length of a particular
ray will clearly be 2s. In (b) the geometry of the diffraction calculation is shown.
The source and image planes are normal to the incident beam direction. For each
scattering angle 6 and azimuth * in the image plane, the total contribution is found
by integrating the contributions of every radius and azimuth angle 4> in the source
plane. This is the familiar Fraunhofer procedure.
decrease gives the total incoherent absorption cross section, simply the
total contribution to 1 —  a  2 integrated over the whole area of the
obstacle. To obtain the coherent, elastic, or diffraction scattering, one
has instead to find the net amplitude which, added to the incident plane
wave in the shadow region, will yield the transmitted amplitude a. This
is plainly 1 — a, and gives a cross section  1 — a  2 .
In the case of a spherical obstacle, the factor a(p) is evidently
(j( n \ iko(B — 2s) (n+ir)2ikos Jko(n — 1) 2s — 2rkos
(117)
with s 2 = R 2 — p 2 . Integrating 1 —  a  2 , we obtain the absorption
cross section:
tfabe = I Pdpi dp(l  e iTSk ") = 2,r I s ds(l  e 2 " 1 )
Jo Jo J
[ 1  (1 + 2R/l)e 2R ' l l
" ' B L 1 2RVP J
(118)
162 A Survey of Nuclear Reactions [Pt. VI
The elastic scattering is rather more complicated. It is given by the
integral (F6) :
•rdiff =fpd P f *d<p\ 1  e *<v<2^i)*o> p (n9)
The angular distribution of the diffraction scattering can be obtained
in a similar way. The approximation here is to sum the contributions
reaching a given image point, far behind the sphere, from every source
point in a plane wave front of amplitude a — 1 within the geometrical
shadow. Then the Kirchhoff integral (of optics) can be written
const
R
ffd<p p dp e ik ° R  pA B X[1 a(s)] (120)
where the geometrical relations are those of Fig. 29. Using polar co
ordinates in the effective source and image planes, we can write
Rv
i r = p(cos <p cos $ + sin <p sin $) X sin @
and the entire integral becomes
"p dp d<p e ik ° 8in e cos ( **V(s)  1]
If
Using a familiar representation for the Bessel function, we can write (F6)
p dp J (k p sin @)[1 — a(s)] (121a)
J
•'o
and the differential scattering cross section dc/dil is just proportional to
I «ac  2  We can evaluate the constant in the limiting case of R/l —+ <».
For a totally black sphere, this goes over to the case already given in
Eq. (46), namely: da/dti = R^J^koR sin 6)/sin @] 2 . For the gray
spheres here under consideration, the evaluation can be carried out.
By replacing the integral over p by the sum over I, in which I + ^ = kp
and the relation Jq[(1 + 2) s * n ®1 = Pi( cos ©)> valid for large I and small
@, is employed, we find
l + V 2 <kB
a sc ~ E (21 + l)Pi(cos »)[1  a(si)] with
si = [(k R) 2 (1 + \fV A IK (121b)
This is very reminiscent of the partialwave analysis of scattering, and
indeed is just the W.K.B. approximation to the phase shifts of the
Rayleigh partial wave procedure. That procedure has been used ex
Sec. 11C] Nuclear Reactions at High Energy
163
actly (PI), giving a result somewhat larger than does (121b) at small
angles of scattering. A series procedure is also applicable for the evalua
tion of (121a) directly.
The results of such a calculation are indicated in Fig. 30. The effect
of nuclear transparency is to reduce the contribution of the shadow
scattering as the transparency increases (or R/l decreases). The graphs
show this reduction in the case of a plausible value for the index of re
fraction, implying some increase in phase velocity within the nucleus.
2.0 
03
J= 1.0
^
M=1.16
"totA^ 2
<W*R 2
°abs/Tfl 2
1.0
R/l
2.0
Fig. 30. Cross sections computed from the optical model. The diffraction, absorp
tion, and total cross sections, measured in units of the geometrical cross section,
are given as a function of the nuclear radius measured in units of the free path. A
particular value, n = 1.16, has been chosen for the refractive index. From (F5).
When n is not simply unity, the diffraction scattering may considerably
exceed the geometrical limit wR 2 for a black nucleus.
By fitting the data on neutron scattering at 90 Mev to the results of
Eqs. (118) and (119), we can choose n + ir in such a way that the value
of R obtained for a certain choice of n + ir fits for all nuclei from Li to U
the simple relation R = r A H , with r chosen at 1.39 X 10 13 cm, a
very reasonable value. Suitable values, which are consistent with direct
scattering experiments for neutronproton and protonproton scattering
at this energy are I = 3.3 X 10 13 cm and n = 1.16 (F5). For these
values the coherent scattering is larger than the geometrical value.
A less empirical evaluation of n + ir is possible in two ways. The
nuclear material simply acts to change the phase velocity of the waves,
and can thus be represented by a potential well of depth V, with the
relation n = k/k = (1 + V/E) y% . This works well enough for 90Mev
neutrons, but the same choice by no means satisfies the experiments at
higher energy, where the variation of total cross section with energy
does not follow the predictions of the simple potential well idea. A
much more sophisticated approach has been made (J2) by means of the
164
A Survey of Nuclear Reactions
[Pt. VI
familiar result of optical dispersion theory, which gives the refractive
index in terms of the scattering amplitudes for forward scattering by the
individual nucleons. The relation is
n  1 = K„(0) + a n . p (0)]Trp/k 2 (122)
where p is the density of nucleons in the nucleus, and the a's are the
appropriate scattering amplitudes. Use of the empirical cross sections
da/dil
barns
24° 16° 8° 0°
Neutron scattering angle
Fig. 31. Angular distributions of elastic scattering computed from the optical
model. The differential cross section is shown as a function of scattering angle for
two nuclear radii (corresponding to copper and to lead). The experimental points
are shown. All this is for 90Mev neutrons incident. From (F5) and (CIO).
is required for I, .but it is of course not enough to fix the quantities a in
(122) unambiguously. Best fit with the observed total cross sections for
a nucleus like Al is found when the real part of the index of refraction is
made to go rapidly to unity, k/k — > 1 — as the energy increases above
100 Mev. Such a scattering amplitude results from a nucleonnucleon
potential consisting of a strong repulsive core surrounded by an attrac
tive well, with a core radius of some 10 13 cm. This result can be taken
as some confirmation of the view, based on the very flat and strongly
isotropic protonproton cross section above 100 Mev, that some such
"hardcore" model of the nucleonnucleon interaction is at least phe
nomenologically suitable. With a n . v = a p . p the fit for the total cross
section to relation (119) is quite good.
The angular distributions are another source of information. In Fig.
31 we show a comparison between theory and experiment for neutrons
Sec. 11D] Nuclear Reactions at High Energy 165
of about 90 Mev. The diffraction pattern is qualitatively correct, but
the experimental peaks are a little sharper, as though the nucleus were
somewhat smaller than expected from the constantdensity model.
Experiments at higher energy show the positions of the successive
minima in the diffraction pattern, and indicate that the minima are in
fact blurred out, perhaps only slightly in the heaviest nuclei, but beyond
recognition in the lightest ones. This is consistent with the results to
be expected if we surrender the naive idea of a strictly uniform sphere of
nuclear matter and imagine instead that the nucleus has a core of con
stant density surrounded by a fringe of nuclear matter with a thickness
something like the range of nuclear forces within which the density
falls gradually to zero. Such a model is of course actually implied by
the simple notion of a nuclear surface, as it is for any classical drop of
liquid. Taking such a model, and the connection between n and the
empirical nucleonnucleon cross sections given by the dispersion theory,
Jastrow (J3) claims a reasonable fit, both to the total cross sections and
to the details of the angular distribution of the diffraction scattering,
over a wide range of energy and nuclear mass. Much more remains to
be done.
D. Processes of Nuclear Deexcitation at High Energy
Let us return to the topic first mentioned in this chapter: the detailed
course of nuclear disintegration in the region of hundredMev energies.
In the stepbystep treatment of the cascade, each particle was followed
until the kinetic energy it possessed, relative to the bottom of the over
all nuclear potential well, fell below a certain limit, some 30 Mev. After
this, no single collision between this particle and the nuclear edge is
likely to result in the emission of the particle; rather, reflection will
occur, and the energy will gradually distribute over the whole of the
nuclear system. Not until it is again by chance concentrated on a single
nucleon (or in a single welldefined mode of nuclear motion) will emission
occur. This is the basis of the compound nucleus treatment which is the
main burden of this entire discussion. To place matters in terms of the
familiar thermal analogy: excitation of a nucleus by collision with a
single nucleon, or a small nucleus, corresponds to heating the nuclear
matter very hot indeed in a small spatial region. From this region a
nucleon or several may emerge, taking off most of the energy. But
the heat will proceed to spread. The time of conduction of energy over
the whole, compared with the time it takes a single nucleon or a few
nucleons of a cascade to emerge, will determine whether or not the subse
quent steps will proceed by thermal evaporation. Our treatment
assumes that, after a certain energy loss has been suffered by each fast
166
A Survey of Nuclear Reactions
[Pt. VI
recoiling nucleon, all the remaining excitation energy passes into thermal
form. Since the spread of energy is mainly through the zeropoint
motion of the Fermi gas of nucleons, this implies that the statistical
treatment will be valuable when the individual nucleon has an excitation
not large compared to its zeropoint energy, well under 20 Mev or so.
This limits the process plainly to cases where the whole nucleus is excited
by a good deal less than its total binding energy, say by some 5A Mev at
the most. We have already seen that such excitations are ample to
handle the most frequent cases observed in the bombardments with
Bi(p, 3n)
10 20
Energy of incident protons (Mev)
Fig. 32. Reaction cross sections for various protoninduced reactions on bismuth.
The competition between the several modes of decay is shown strikingly here; as
each new mode becomes energetically possible, it tends to drain away the previously
favored decay method. After (K5).
nucleons up to almost 500 Mev in fairly heavy targets, and they cover
also, at least in not too small nuclei, the interesting phenomena following
upon meson absorption, both w and n (B12, M6).
The statistical treatment we have used earlier in the lowenergy cases
implies of course a thermal equilibrium, which will not actually exist
in the case of high excitation and the emission of many particles. We
can expect that a kind of transient equilibrium will in most cases exist
between the successive nucleon emissions which make up the total event.
The limit on such an idea is no different from that mentioned just above:
the time for rediffusion of the excitation energy after an act of emission,
governed by the thermal transport, should be small compared to the
delay before the next concentration of energy on a single emitted particle.
The typical thermal transport time is under 10 21 sec; while the time for
nucleon emission up to the highest excitations here considered is ten
thousand times as great. The whole division of the process into an
immediate cascade and a subsequent evaporation is of course approxi
Sec. 11D] Nuclear Reactions at High Energy 167
mate; but within these limits the approximation is successful to a rather
high degree, and very instructive.
Of course, the transition from the ordinary statistical treatment of
reactions like (n,a) or (n,p) to the large evaporative stars so dramatically
revealed in the nuclear emulsion studies is a continuous one. For in
stance, an elegant series of experiments by Kelly (K6) has carried the
simple competition theory of formation of the compound nucleus and
its successive decay (see Section 7) through the whole series of reactions
with the target nucleus Bi 209 : (p,n), (p,2ri), (p,3n), (p,4n). Excitation
energies in this experiment ranged up to about 40 Mev. Figure 32 shows
the results obtained, which fit the expected relations of the statistical
theory of competition [see Eq. (89)] very well indeed. If we write the
ratio of excess energy of excitation over threshold to the temperature
as x, the data fit the simple result a{p,2n)/a v = 1 — (1 + x)e~ x very
well by adjusting the single parameter, the temperature kT. The value
chosen for kT agrees completely with the calculated value for the semi
empirical level density formula, Eq. (31), about 1.1 Mev.
1. Spallation and Evaporation Stars. As the bombarding, and hence
in general the excitation energy, continues to rise, we reach a region of
higher complexity. Here not only a single product, or a few related
competitors, can emerge, but a whole series of successive steps can be
taken, each one in many alternative ways. This fact, and the often
important fluctuations away from the mean statistical behavior of even
such a complex multiple evaporation, is the reason for the absence of
any wellcodified study of this general subject. So far we can give fairly
satisfactory but rather tedious means of calculation, but no compact
summary of the possible results.
Two different experimental approaches to the field have led to different
nomenclature, and to somewhat different descriptions, of one and the
same phenomenon. The radiochemical procedures of separating the
various radioactive products from targets bombarded with projectiles
at differing energies, and so obtaining yield curves for a whole series of
possible products, is one powerful means of study (B4). Here of course
only the overall change is observed; if A and Z change by 20 and 10
say, the event can be described as the emission of 10 neutrons and
10 protons. But the emission of alphas or even other small nuclear
fragments in such an event is by no means excluded. Typically a high
energy bombardment yields a bewilderingly large number of product
nuclear species. By extension from the idea of the fission of a nucleus
into two large fragments, with a little neutron spray, the radiochemists
have come to call these processes spallation, with the implication of the
168 A Survey of Nuclear Reactions [Pt. VI
emission of a whole series of small fragments, rather than the cleavage
into two big ones familiar in fission.
On the other hand, the cosmicray workers, first with their cloud cham
bers (P4), and more recently with the powerful nuclear emulsion tech
niques (B9, H3), have typically studied not the statistical residue of
millions of disintegrations by identification of the product nuclei, but
the ionizing fragments released from each individual nuclear breakup,
with the familiar bristle of ionized rays, each marking the trajectory of a
single ionizing fragment starting in the central decaying nucleus and
ending at the end of the fragment's range. In this method, besides the
simple count of the fragments and charge carried off, the energy and
angular distribution can also be obtained. But a good statistical sample
is tedious to collect, and neutrons are invariably missed. (A few ex
periments with counters have satisfied us that the invisible neutrons,
expected in more or less equal numbers with the visible "prongs," are
really emitted.) The typical appearance of the event in the emulsion
has led to its designation as a "nuclear star." Star or spallation reaction,
the event is the same, but the method of detection, and to some extent
the features used for description, differ widely.
In Fig. 33 we present a rather generalized summary of the results of a
typical spallation study: the bombardment of a copper target with
protons of 340 Mev (B4). The contours in the N,Z plane show the ob
served yields, with some plausible extrapolations and smoothings.
Characteristic is the large number of observed nuclear species, more
than thirty having been identified. The biggest yield is for nuclei
differing by only a few units in Z and N, plainly due to events in which
not much excitation was left behind after a small cascade. There is a
strong tendency to stay in the vicinity of the stable valley; it is not
likely that the neutronproton balance will be badly upset by evaporative
events in which nearly all the emitted nucleons are protons (or neutrons).
This tendency is compelled by the quasiequilibrium theory, and favored
by the energetically advantageous emission of alphaparticles themselves.
The spallation studies have shown one other rather interesting phe
nomenon (B5). Both by identification of the shortlived isotope Li 8 ,
and by arguments of energy conservation, it has been demonstrated
that at least a few events are possible which bridge the gap between
simple alphaemission and straightforward fission. In Section 10, it
was pointed out that the splittingoff of small fragments is energetically
favorable for a wide range of A because of the balance of Coulomb and
surface energy. In spallation experiments, where infrequent events
can be detected with relative ease, it has been found that there is indeed
a small yield of nuclei of mass far removed from that of the target even
Sec. 11D] Nuclear Reactions at High Energy
169
for rather low bombardment energies. For example, 70Mev protons
on Cu give a small yield of CI 38 . The connecting reaction which in
volves emission of single nucleons and alphas only is the reaction
Cu 63 (p,6apn)Cl 38 . But that reaction has a threshold, calculated from
mass differences, of some 110 Mev. The extreme reaction, from the
Sq25 
Fig. 33. A contour map plotting the yield of a spallation reaction in the N,Z plane.
The target nuclei are marked with solid circles; copper was bombarded with 340
Mev protons (B5). The yields center roughly on the line of greatest stability, shown
dashed.
point of view of minimizing surface energy, is the threeparticle breakup,
Cu 63 (p,nAl 25 )Cl 38 . The threshold for this rather unlikely course of
reaction is about 50 Mev. It thus seems established that a variety of
heavier fragments, of which Li 8 has surely been found, but probably
extending over a wide range of masses, up to processes splitting the
nucleus into several pieces of approximately equal weight, can in fact
take place. This marks a kind of transition to the case of lowenergy
fission, and implies that the statistical competition of specialized vibra
tions of the nucleus leading to massive splits will have to be taken into
account in a complete theory of cooling.
170 A Survey of Nuclear Reactions [Pt. VI
2. Evaporative Stars: The Fundamental Process. The study of
the energy and angular distribution of observed star prongs enables a
rough distinction — allowing considerable overlap — between the cascade
and its evaporative consequences. The evaporative prongs are gen
erally marked by: (1) a roughly isotropic angular distribution, (2) a
rapid decrease in prong frequency as energy increases. The long tail
up to high energies actually observed — socalled "gray" and "sparse
black" prongs, named from their low ionization densities in the emulsion
(B9) — are regarded as cascade particles. True evaporative particles
ought to show a typically quasiMaxwellian dropoff of number with
energy, and no such long tail to energies far beyond any possible mean
particle energy. In general, we expect the evaporative process to be
sensitive, especially in its last stages, to details of nuclear structure, like
the oddeven property and even more refined shelllike behavior.
The thermodynamic treatment of the stars is based on the statistical
formula, due to Weisskopf, which connects the probability for dis
integration of a compound nucleus with the formation cross section, by
the principle of detailed balance. [Compare the discussion in Section 5,
Eqs. (51) and (52).] We write
gM p(f)
2 h 3 p(i)
where P(T) dT is the probability per unit time for emission of a particle
of mass M and spin statistical weight g with kinetic energy in dT; a is
the cross section for capture of the same particle by the residual nucleus /
to form the initial nucleus i; and the functions p are the level densities
of initial and final nucleus as indicated, functions of course of the mass
and charge number and of the initial excitation energy, W, and final
excitation W.
In principle, we may now use the function Pj(T) for emission of a
particle of type j with mass A 3  and charge Z }  from the initial nucleus of
given kind and given excitation energy, W = W + T + Bj, with B } 
the binding energy, and then simply follow the course of the successive
emissions until all the excitation energy has been converted into kinetic
and potential energy of the fragments, and into radiation. This implies
the solution of the set of coupled equations which can be given, if only
formally, as (F13)
dy(Z,A,W,t) „_ r x
' = E dW'y(Z + Z h A + AjWdPjiW'.W'  W)
dt j JW+Bj
P(T)dT = ^— ff (T)T — dT (123)
 Z y(Z,A,W,t) f P 3 {W,T) dT (124a)
Sec. 1 ID] Nuclear Reactions at High Energy 171
Here the yield function y(Z,A,W,t) gives the probability of finding a
nucleus with charge Z, mass A, excitation energy W, at time t after the
start of the process, which fixes the initial condition:
y(Z,A,W,t) = Saa, 5zz„ S(W  Wo) (124b)
Now from the function y one can learn all the facts about the star,
including of course both the mean behavior and fluctuations from it.
For example, the total yield of a definite nuclear species is just given by
Vtot(4,Z) = f 'y(Z,A,W,t » oo) dW
Naturally such a complete treatment is next to impossible; indeed,
fluctuations are generally treated very casually [but see (Tl)]. We
shall examine a more manageable approximation; perhaps the complete
treatment will one day be performed by Monte Carlo technique, which
seems very well suited to the task.
3. The Mean Behavior in Cooling. The most satisfactory approach
is the study of the mean behavior of the evaporative process. To make
this possible in general (F13), several approximations must be made,
not all of easily controlled accuracy:
1. In the fundamental formula (123), the cross sections are replaced
by very simple forms: usually just the geometrical irR 2 for neutron emis
sion, and a simplified expression like <r = wR 2 (l — V'/T) for a charged
particle with a Coulomb barrier height given by V = 0.7(ZZje 2 /R), to
take care of penetration effects.
2. The level density is expressed by some one of the results of a par
ticular nuclear model, discussed in Section 3. The detail with which the
shell properties are represented may vary greatly.
3. The cooling proceeds usually by the emission of particles whose
Aj and Zj are small compared to those of the initial and residual nucleus.
This allows series expansion of functions of A and Z, performed as
though A and Z were continuous variables, instead of integers.
First of all we define the relative width (hence probability) for emitting
a particle of a given kind, indexed by j. In the typical calculation, it
may be worth while to consider halfadozen or more types of emitted
particle, as proton, neutron, deuteron, H 3 or triton, He 3 , He 4 or alpha
particle, and sometimes even heavier fragments. In general, then, the
mean emission width is given by the expression
Tj(W) = f Pj(W,T) dT
JVf+Bj
where Pj is the transition rate given by (123). The approximation of
172 A Survey of Nuclear Reactions [Pt. VI
mean cooling is of course just to consider the process as going always
proportional to these mean emission rates, and to neglect the different
courses of the evaporation which come from the fluctuations about the
mean energy and type of decay. Naturally no single act follows the
average behavior, and the successive steps in fact depend in turn upon
just what particle and energy loss preceded them. These refinements
are here neglected. Now we can calculate the mean energy of excitation
lost by evaporation of particle j:
X
Pj(T)TdT
8Wj = Tj + Bj = + Bi (125a)
X
PjdT
and we can express the mean cooling in one way by writing the mean
rate of energy loss per nucleon emitted :
— Z IV HWj
SW j
■= = (125b)
8A Sr, 5Aj
In a similar way we can write the mean number of particles of type j
lost per unit energy loss :
Srij Tj
= = = (125c)
8W HTjdWj
and for the entire cooling from initial to final excitation energy get
— r w * Tj{W) dW
dnj =  ^ (125d)
■ _ r w < Tj{W) dW
' 3 Jvj'+bj ]r Tj s~Wj
Here there is of course one such equation for each of the values of j, say
six or more, and the equations are all coupled. The main coupling is
through the effect of the changing neutronproton ratio, and some
reasonably trustworthy solutions have been obtained (neglecting other
relations) (L3).
(a) Cooling Behavior with an Explicit Model. To gain any insight
into the expected behavior of this phenomenon, it is necessary to intro
duce an explicit nuclear model which can fix the level density and its
variation with Z,A, excitation energy, and even finer details like the
oddeven effects. In Section 3 we set out a number of expressions from
various models for the level density p(W). We choose here to use the
rather simple model of the Fermi gas, which has some weak experimental
Sec. 11D] Nuclear Reactions at High Energy 173
support, and can be regarded as the empirical expression of the results
for nuclei of middle weight and for energies in the range here studied.
With a Fermi energy of 22 Mev, we obtain [compare Eq. (31)]
^(.A.Z.W) (AW)**
P (W) = v and S = 0.63 (126)
P ^ ; t(2t dW/dr)* Mev
Here the entropy is given as a function of the excitation energy W, and
we have introduced the temperature t, writing W/Mev = At 2 /10, and
t is in Mev. With this model the rate of evaporation becomes, neglecting
some slowly varying ratios,
P(T) dT = gM/ir 2 h 3 (T  V')e^~ Si) dT (127)
and, if we make the approximation, valid for small emitted fragments
and 8W « W,, of expanding the function S(A,W), we get S(A — SAj,
W  8W) = S A (W)  SW/t  SAjt/10 and
~(T  V)
P{T) dT = const (T  V) exp (128a)
T
This is the familiar Maxwellian distribution of evaporated fragment
energies. The mean kinetic energy taken off by an evaporated particle
i is
I x 2 e x dx
Tj = t — = 2r (128b)
3 ,00
xe~ x dx
I
and the total width T 3 {W) for emission of jparticles with any energy
T is approximately
Tj = h P(t,W) dW
JBj + Y'
~ a — i ■ exp I 1 t in Mev (129)
~ y 2x 11 Mev V 10 t /
Both terms in the exponent here depend strongly on the type of
particle emitted; they are the effective potential barrier V/ and the
binding energy Bj. Now we must explicitly give the variation of
binding energy with A and Z, which will of course determine the nature
of the most probable emitted particle. We write a semiempirical form
for the binding energy of a nucleus in its ground state [compare Eq.
174 A Survey of Nuclear Reactions [Pt. VI
(101)], with <(JV  Z)/A ) written for the value of (N  Z)/A which
corresponds to the most stable nucleus of a given A :
B = const + Cl A  c 2 A[(N  Z)/A  { (N  Z)/A >] 2
Here we can take c x about 8.6 Mev, and c 2 about 23 Mev. These num
bers are suited for target nuclei in the middle of the table, A ~ 100, as
in photographic emulsion. Now, if a particle of type j is evaporated,
with 8n,j neutrons and SZj protons, taking off kinetic energy T, the drop
in excitation is given by differentiating B, and regrouping the terms,
to yield
8W = T + Bj
[N  Z IN  Z\l
Bj = Cl dAj  2 — \~J~ / J < c " bn * ~ c * SZ ^ + 7 >
f a IN  Z\ l\NZ IN  Z\l)
c " =C2 { 1  A M\i/i[x + \x)JH a8c2
f a IN  Z\ 1VNZ IN  z\\)
(130)
and Ij is the internal binding energy of the emitted fragment. The
energy Bj is of course the threshold energy for emission of the given
fragment with zero kinetic energy, which we call the binding energy of
the fragment into the initial nucleus. We have considered only nuclei
rather near the stable valley — (N — Z)/A ~ {{N — Z)/A ) — in evalu
ating c n and c z .
Now we can examine at least the general cooling behavior by simply
looking at the competition between the various emitted fragments.
Q>) The Competition in Cooling. Let us look at the ratios of the r,
for the various particles. First, note that the statistical and mass factors
g 5Aj alone will give relative weights for the six most common products
as follows: p:n:d:T:He 3 :a: :1:1:3:3:3:2. Note the increase in statis
tical probability, especially for the heavy isotopes of hydrogen, which
feel a low Coulomb barrier as well.
Much more important, however, than the statistical and mass factors
is the exponent in Eq. (129). We write here the expected emission width
relative to that for neutron emission, using the energy relations of (130) .
We can write the effective Coulomb barrier heights for nuclei of the
emulsion (A ~ 90) as V/ = 6bj Mev, where the values of bj are esti
mated to include charge, radius, and mass effects on penetrability, and
Sec. 11D] Nuclear Reactions at High Energy 175
we get b n = 0; b p = 0.7; b^ = 0.8 = br; &He 3 = 16. Then the widths,
relative to the width for neutron emission, become
r„/r„ = l
iyr„ = exp [ (92(i. v) + 4)/r]
T d /V n = 3 exp [t/10  (56(i  v) + ll)/r]
T T /T n = 3 exp [2t/10  (18^  v) + 14) /r]
r H ea/r n = 3 exp [2t/10  (110(?  r) + 19)/t]
TjT n = 2 exp [3t/10  (74(*  p) + 8)/r]
The energies involved and the temperature r are all given in Mev. We
have written v for the quantity (N — Z)/A ; the value of v for the most
stable nucleus of a given A is written as V. Thus {v — v)A = neutron
excess — proton excess.
An examination of these ratios alone enables a qualitative discussion
of the course of the mean curve of cooling. The most noteworthy
features are these:
1. The Coulomb barrier — contributing to the last term of the ex
ponent — of course favors the emission of neutrons over charged particles,
and reduces the emission of particles of charge +2, as long as it is of
any consequence. Since the temperature r is ~(W)**, the effect of the
Coulomb barrier on chargedparticle emission will be decisive for all
excitations up to some critical energy. For higher energies the emission
of ions and neutrons will show little difference ascribable to Coulomb
effects. The critical energy for proton barrier effect in these middle
nuclei is an excitation of from 100 to 150 Mev; for helium isotopes,
from 250 to 350 Mev.
2. Apart from the Coulomb barrier, the most striking term in the
exponent of the relative emission widths is the symmetry energy, propor
tional to the fractional neutron excess of the initial nucleus, (v — v).
Looking, for example, at the protonneutron ratio, T p /T n , we see that
this term favors neutron emission when there are too many neutrons
compared to the region of stability, and favors proton emission when
the nucleus is neutron deficient. This term has been called a "governor"
term (L3), because it prevents great excursions from the region of stable
nuclei in the cooling process. It arises of course from the parabolic
form of the familiar Heisenberg valley in the nuclear energy surface.
In manyparticle stars it will more or less insure that more neutrons
than protons are emitted.
3. The leading term in the exponent of the relative widths for emission
of the complex star fragments — d, a, etc. — grows more negative as the
176
A Survey of Nuclear Reactions
[Pt. VI
excitation, and with it the temperature, increases. This term arises
from the dependence, not of the energy, but of the entropy itself, on the
number of nucleons in the nucleus. It favors the emission of lighter
rather than heavier particles, because more ways exist of assembling
a nucleus if it contains more nucleons. If it were not for this type of
term, the emission of the heavier fragments would become more prob
able than that of single nucleons because of the purely statistical weights.
Fig. 34. The course of the neutron number of an evaporating nucleus as a function
of nuclear temperature. The neutron excess is plotted; it is the difference between
the number of neutrons in the nucleus, N, and the number of neutrons characteristic
of the most stable nucleus of the same mass number A. The neutron excess first
increases and then decreases as the nucleus cools. The terminal value of N — (iV)
is about —1, implying the subsequent emission of a slow proton, or simply /3 + decay,
following the main process of evaporation. See pages 174177. The curves are
drawn for a nucleus with initial A = 100, and for two initial temperatures, 4 and 8
Mev. After (L3).
For example, without this term r a /T n — > 2 as t grows without limit.
As star size increased, then, alphaemission would outweigh proton
emission and even neutron emission, a very odd result, and far from what
is actually observed. With this entropy term, however, the ratio
Ta/Tn goes through a very flat maximum in the region of 100Mev
excitation. As we have seen, however, in earlier chapters, the variation
of level density with A is quite poorly known, and detailed predictions
from this simple entropy assumption cannot be expected to agree in
detail with experience.
Now we can integrate the simultaneous equations (125) to give the full
course of the cooling. This has been done in rather rough approxima
tions, perhaps best by LeCouteur, whose procedure we follow (L3).
Sec. 11D] Nuclear Reactions at High Energy
177
In general, we can expect this typical state of affairs : the highly excited
nucleus, say with 400 Mev, begins by evaporating neutrons and protons
about equally. To begin, it has v — v almost zero — perhaps one or two
protons and neutrons have been knocked out of the stable target nucleus
in the nuclear cascade. But now the nucleus boils off several protons
200 400 600
Initial excitation energy (Mev)
Fig. 35. The number of evaporated particles of several types as a function of initial
excitation energy, for an evaporating nucleus of initial A = 100. The neutron num
ber scale is at the left; that for chargedparticle emission at the right. After (L3).
and several neutrons. By the time it has emitted say four of each, the
cooling nucleus has become neutronrich, with respect to the smoothed
over line of maximum stability, and the "governor" term begins to
favor the loss of neutrons. Then the cooling proceeds, somewhat prefer
entially losing neutrons, until the excitation has dropped below some
100 Mev. From here on, proton loss will be almost prohibited by virtue
of the Coulomb barrier, and the excitation energy will fall until no more
particles can be emitted. The resultant nucleus is now neutrondeficient
by a little less than one neutron on the average, more or less independent
of the initial energy of excitation for not too low excitations. We show
in Fig. 34 a schematic plot of the neutron number against excitation
during the cooling, which follows the course here described.
178
A Survey of Nuclear Reactions
[Pt. VI
In Fig. 35, we plot, after LeCouteur, the mean numbers of various
emitted particles for different excitation energies, with an initial nucleus
of A — 100. The virtual independence of the relative yields of various
particles upon energy is a feature of the theory; but the calculations have
actually been done neglecting the small variations actually predicted,
10 20 30
Kinetic energy of evaporated particle (Mev)
Fig. 36. Energy spectrum of evaporated particles for several initial excitation
energies. Initial nucleus of A = 100. The solid curves all refer to emission of pro
tons; the dotted curve, to that of alphaparticles. After (L3).
and using the relative yields given by equations for an initial excitation
of 400 Mev. In Fig. 36 we give the expected kinetic energy distribution
of evaporated fragments; it is given by the rough averaged formula,
very much as in (127), by
P(T) AT =
V
(TV)/t
dT
(132)
Results are given for several excitation energies, and for both singly
charged and doubly charged star prongs.
Sec. 11D] Nuclear Reactions at High Energy 179
All these results are likely to be acceptable only for fairly large excita
tions (W > 150 Mev), and the theory cannot be taken seriously for A
much less than about 60. The actual figures are for a definite A = 100.
(c) The End of the Cooling Process: LowEnergy Prongs. We followed
the cooling process down to the point where the temperature is so low
that no further nucleon emission is possible energetically. But this
threshold of course varies with the type of particle. Moreover, the
process has left us with a nucleus which is neutronpoor by about 0.8
neutron. Then the values of the threshold energy are not the same as
for a stable nucleus, and become, from (130), B p c^7 Mev, B n ~ 10 Mev,
5 a ~ 5 Mev for the three main possibilities. Thus, even though the
Coulomb barrier reduces the rate of chargedparticle emission for such
low energies very much indeed, the neutron cannot compete at all,
since the binding energy of the neutron in the neutronpoor residual
nucleus is higher than normal. Deexcitation can go on by gamma
emission, betadecay, or chargedparticle emission. The anticipated
lifetime for these slow processes at an excitation of 10 Mev is perhaps
10~ 16 sec or so for the gamma, and some 10 2 sector the beta. Protons
emitted through the barrier with kinetic energies of anything above
0.5 Mev will favorably compete with such slow radiative processes.
Alphas and other fragments of mass above one unit will experience a
much greater loss in penetrability; since this factor is already 10 10 ,
protons are about the only practical competitors. Thus, if a nuclear
particle of any kind, the nexttothelast to be emitted, leaves the residual
nucleus with an excitation of less than 10 Mev and of more than B p + 0.5
or some 7 Mev, a proton will be emitted. If the remaining excitation is
below B p , gammarays alone can come off. Thus about 30 percent of all
nuclei left with 10 Mev excitation will emit slow protons, far more than
would be expected if neutron competition were not excluded by the shift
in binding energies due to the neutron deficit. A computation (L3) has
shown that there is to be expected in emulsion nuclei about 0.2 proton
per star with kinetic energies in the range from 0.5 to ~3 Mev. These
very slow protons have apparently been observed (H3). In addition
there is a considerable excess of slow alphas observed; this cannot be
accounted for by such a mechanism, but is ascribed to a lowering of the
effective barrier by processes to be considered in Section HE (L3).
It will not escape the reader that this type of calculation has over
looked even the most striking features of nuclear shell structure, like
the oddeven variation in energy content. This does not affect the
previous phenomenon, slowproton emission, but in general it should
have some detailed effects on the closing scenes of the evaporation, when
excitations are not large compared with the few Mev which represent
180 A Survey of Nuclear Reactions [Pt. VI
shellclosing energies, etc. Moreover, the rather large gammaray width
which seems to be observed in the neighborhood of 20 Mev excitation
may occasionally compete — say one time in ten or a hundred — with
the last one or two emitted particles. All these interesting but somewhat
fine points await both more systematic experiment and a better theoreti
cal treatment.
(d) Effects of High Nuclear Temperature. The calculations of evapora
tion ought not to neglect the changes in the nuclear properties which
arise from the high nuclear temperatures during the earlier stages of the
process. In equation (126) and in the bindingenergy formulas of (130)
we have oversimply considered the nuclei throughout as though they
were in their ground state. The work of LeCouteur (see Figs. 34 and
35 which are essentially taken from his work) actually did not leave
out this important and complicated circumstance. The effects to be
considered are three in number, all closely related to thermal expansion:
1. Depression of the Coulomb barrier, and consequent favoring of
emission of lowenergy charged particles, as a result of the thermal
expansion of the nucleus. An estimate of the thermal expansion of the
nucleus due to an actual change in the equilibrium volume energy at
high temperature, entirely analogous to the familiar phenomenon on
large scale, can be obtained by thermodynamic arguments. We can
write the internal energy of a nucleus of given A as U(V,t), showing its
dependence on temperature and volume. For a definite temperature —
actually t = — this internal energy is a minimum at some volume, say
V . Expanding, we can write
dU
U(r,V) = U(t,V ) + —
(VV ) 2 +
But the coefficient of the linear term is zero by the definition of V 
Now the Helmholtz free energy, for negligible external pressure, is just
F = U  rS = U(r,V )  tS(t,V) + 
(v  v y
ld 2 U
2~dV 2
and the actual equilibrium volume at any temperature r will satisfy
= fixing the free energy at a minimum. We have
dF
the relation — ■
dV
already given the entropy in terms of A (Eq. 126) and we can estimate
the entropy as a function of volume by using the relation V = ^irr 3 A.
This of course is not precise; the entropy depends both on A explicitly,
the number of nucleons, and on the volume for fixed A, but it will serve
Sec. 11D] Nuclear Reactions at High Energy 181
for an estimate. Now we get S = 0.2At = 0.2 • F/(4Trr 3 /3) • t, and this
yields an expression for the volume change with temperature:
V = V + [0.2/(W/3)] • [r 2 /(d 2 U/dV 2 \ TjVo )] (133)
The familiar thermodynamic result dF = —S dr — p dV for small depar
tures from equilibrium yields the relation dF/dF T = —p. But the ordi
nary isothermal compressibility is defined as k = —(1/V) (dV/dp)\ T .
Since dp/dV\ T = l/(dV/dp)\ T , we have l/« = Vd 2 F/dV 2 \ T . For low
temperatures we can write, closely enough, d 2 F/3V 2  T = d 2 U/dV 2 \ T .
An estimate of the nuclear compressibility, d 2 U/dV 2 , is easy to
obtain in several ways. An application of the virial theorem (78)
gives d 2 U/dV 2 = fc£7 kin /F 2 , where the factor k is about 2; direct calcu
lations with models using a Fermi gas and correcting for assumed two
body interaction potentials are in fair accord. Some insight may be
had by recalling that the velocity of sound v is given by the relation
w 2 = [V(d 2 U/dV 2 )]/p. If the sound velocity is estimated very
roughly as simply the rms velocity of the particles in the Fermi gas, we
have v 2 /c 2 = 6E F /5Mc 2 . Using (133), we can write the relative change
in radius upon heating [R(t)  R(0)]/R(0) ^ %(r 2 /E F ) = 0.008t 2
(t in Mev), a small but not entirely negligible change. Bagge (Bl)
has pointed out that there is likely to be another, more dynamical,
source of reduction of the Coulomb barrier than this uniform volume
expansion. Surface vibrations in the excited nuclear drop "wrinkle"
the surface; over many cycles, protons in the nuclear surface layers are
to be found farther out from the center than in the smooth undisturbed
drop at zero temperature. The diagram of Fig. 37 shows the nuclear
a
Q
n~\
OR OR OR
(a) (b) (c)
Fig. 37. Schematic plot of density of nuclear matter in a crosssectional cut through
a nucleus. In (a) is shown the abstract model of a uniform sphere without surface
effects; in (b), the more realistic version of the Fermi model, with a fringe of grad
ually falling nucleon density; in (c), the nuclear density in a [highly excited and hence
much expanded and strongly oscillating nucleus.
density distribution at zero excitation and at high temperature; there
is both a lowering of the mean p and a wide fringe of nuclear matter as t
increases. An estimate of a 2 , the mean square amplitude of the surface
vibration normal modes at a given temperature, has been made by Bagge,
who shows that although the number of modes excited is not large, and
182 A Survey of Nuclear Reactions [Pt. VI
quantum treatment is necessary, the value of a 2 is nevertheless very
closely proportional simply to the nuclear temperature. This can be
described by saying that the surface vibration energy is simply a definite
fraction of the total excitation (F14) and that the mean frequency of
the modes excited changes slowly. The surface effect appears to be quite
important, reducing the barrier in the ratio V'/V — 1/(1 + "^ a 2 /R),
and a semiempirical estimate of the effective proton barrier as V p '
= 4/(1 + 0.15r) seems to fit the present data reasonably well. With
this variation of barrier with temperature, the spectrum of emitted
charged particles of course changes, and this reduction of V is fairly
surely observed (Bll, B8).
2. The excited nucleus has a different balance between protons and
neutrons; at high excitation there is a tendency toward increasing the
difference of neutron and proton number, N — Z. This of course tends
to favor the emission of protons from a nucleus already rich in neutrons.
The tendency originates from a complex set of causes, most important of
which is a new balance between potential and kinetic energy which has
to be struck for the now expanded volume of the heated nucleus. A
shift in volume decreases the Fermi kinetic energy, and the potential
energy will readjust to fix a new minimum total energy, now at a different
value of v from that of the cold nucleus of the same A. The effect has
been evaluated by LeCouteur most recently, using a Fermi gas plus
interactions. He shows that it leads to an entropy term of the form
+ const [(N — Z)/A] 2 . It becomes appreciable only for excitations
above about 300 Mev. It has only a rather small influence therefore.
It more or less corrects the highenergy emission for the fact that stable
heavy nuclei in the ground state are neutronrich, and equalizes the
protonneutron emission at high excitation.
3. The surface tension energy at high temperature decreases. Evi
dently the nuclear expansion could reach a "critical point" where the
nucleons are all excited beyond their binding energies; here the drop
would vaporize entirely. The nuclear surface tension will decrease
regularly with temperature until the critical temperature, t c , is reached.
Using the rough Fermigaslike relation between excitation and tem
perature, we can write W c = 8A Mev = At c 2 /10, and from this we get
t c = 9 Mev, independent of A in this approximation. Now let us again
take thermodynamics as a guide (G7). If we write down the Gibbs
free energy for a surface film, in which the surface tension —7, surface
force per unit length in the film, takes the place of pressure in the usual
case, we have Gs — U — tS — yA for an area of film A, with internal
energy U and entropy S at temperature r. Now the partial derivative
— 8Gs/dA\ T = y gives the surface tension. An isothermal change in
Sec. 11D] Nuclear Reactions at High Energy
183
film area will obey the first and second laws: dU = rdS + y dA, and
we can write dGs = —Sdr — A dy. Differentiating, — dy/dr = (S/A)
+ (dGs/dr). But, in equilibrium at any r, the Gibbs free energy of film
and bulk liquid are equal for a given mass of nuclear matter. We write
G s = G L . Then dy/dr = (S/A) + (l/A)(dG L /dr) = (S  S L /A).
Now we can form Gs/A = Gl/A = — y + (U/A) — (rS/A), and we
get a differential expression for the surface tension 7 as a function of r :
T +
dy _ (U Ul)
dr A
(134)
The term pV in the liquidphase free energy is cancelled by that of the
surface film, except for thickness changes, which we neglect as usual.
Mev
9
Temperature (Mev)
Fig. 38. The nuclear surface tension as a function of nuclear temperature. The
curves indicate different models: curve (a) arises from the actual surface energy of
a classical liquid drop with surface waves excited by the thermal motion; curve (b)
from the simple linear assumption for the integration of the free energy equation
of the text. The two are very close. From (Yl).
Now the righthand side of (134) is the additional energy due to the
surface film, measured per unit area of nuclear surface. Using as unit
of area just 4tt 2 , this becomes about 14 Mev for nuclei in their ground
states, with r = 0, as the semiempirical formula showed in Section 10.
As the temperature rises, surface vibrations are excited and the energy
increases. Using a liquid drop model, Bethe (B13) has shown that the
excess surface energy increases like r A /y(r). As a first approximation
we can neglect the change in y, and write U — Ul/A = 14 + const t a .
We can then solve (134), using as boundary condition the requirement
that y(t c ) = 0. The surface tension then goes to zero, following the law
7(7") = 14 + t/t c — 15(t/t c Y a . A numerical integration has been per
184 A Survey of Nuclear Reactions [Pt. VI
formed by Yamaguchi (Yl) which gives a very similar result. A plot
is given in Fig. 38.
The reduced surface tension means that the balance between Coulomb
and surface energy is displaced in favor of the Coulomb repulsion. We
would expect this, since the Coulomb forces show only a slow change
with mean separation, while the surface forces are, of course, of very
short range. An expanded nucleus requires little additional energy
to undergo fission, for which process the repulsive Coulomb forces over
come the surface tension; the nuclear volume energy can be taken as
remaining constant throughout. The familiar BohrWheeler estimate
of fission threshold, as given in Eq. (107), is just
E f = iTr 2 A 2A y(r)f
Q
With this model, Fujimoto and Yamaguchi (F12) have estimated that
the silver fission threshold would fall from some 50 Mev for the ground
state of the nucleus to half of that value at about 100Mev excitation,
and to only a few Mev at 350Mev excitation. The fission width can be
estimated in the usual way as about 1/ ~ (r/27r)e~ E//T . This makes
such fission a good competitor for really large excitation, 300 to 400 Mev
or more. Such heavy spallation fragments are found not infrequently,
as we have mentioned above (H3) . Of course, fission yielding fragments
with much smaller mass is even more favored by the decrease in surface
tension, and many reasonably heavy fragments have been seen in high
energy stars, especially the easily recognized unstable ones like Li 8 .
The mechanism for fission here described is only a single one which
may contribute to fission at high excitation energies. With the heaviest
nuclei, and at somewhat lower energies, fission is observed subsequent
to the emission of many neutrons, at temperatures below the barrier
height for protons. This upsets the Coulombsurface energy balance
as well, and reduces the fission threshold. The mechanism is plainly
very different (G4).
It has been observed (F14) that a calculation of the temperature
dependence of the surface energy, using the model for surface vibrations
which Bagge employed, yields not very different results, and still predicts
very easy loss of highly charged fragments at high temperature. Using
Bagge's formulas for the mean amplitude of surface waves, but taking
into account the variation of surface tension with temperature, it appears
that excitations of 300 or 400 Mev would greatly distort the surface
shape of nuclei, so much indeed that whole little drops would come off
in a kind of spray, (a 2 ) l/% /R ^ 1 describes such a condition. It seems
Sec. 11 D] Nuclear Reactions at High Energy 185
rather likely that the actual situation is more nearly a nonequilibrium
local heating, capable of distorting the surface locally enough to pinch
off a small drop, but not involving the surface of the entire nucleus
in such a way as to give the nearly flat distribution of fission frag
ment masses which the neardisappearance of the equilibrium surface
tension might lead one to expect. The whole phenomenon is clearly
of importance, but our present account is too closely classical to trust
quantitatively.
4. Fluctuations in Cooling. We have so far described in detail only
the average behavior of the cooling nucleus; we have ignored all fluc
tuations. Obviously, such a complicated process in which only a few,
or perhaps a few dozen, particles take part must be subject to important
fluctuations in the number, type, energy, and direction of emission of
the emitted nucleons. So far this problem is only sketchily understood,
but a simple model will serve us at least for orientation (F12).
Consider a nucleus so very hot that we can ignore its cooling, even
though it has emitted many nucleons. It is restricted, moreover,
to the emission of only one type of particle, with binding energy B,
which also remains constant throughout the process. Then the prob
ability of emitting a single nucleon in energy range dT is just P(T) dT
— e — T l rr r dT/r 2 , where t is the temperature, taken as constant through
out, and T the kinetic energy. Now the probability that the nucleus
will emit n or more nucleons, when it has an initial total excitation W,
is given by the multiple integral:
P(n,W) =J ■ jdT, ■ ■ dT n _ l P{T i )P{T 2 ) ■ ■ PiT^)
(135)
n— 1
< Z Ti < W  nB
l
where the upper limit follows because there must remain at least B Mev
of excitation after the loss of n — 1 nucleons if at least one more is to be
emitted, and the lower limit represents the process occurring by the
possible route in which each of the first n — 1 nucleons just manages to
dribble out with zero kinetic energy.
It will not seriously distort this rather unrealistic picture to replace
the Maxwell distribution in kinetic energy by a Gaussian distribution
adjusted to have a mean and a standard deviation equal to the Max
wellian values. We know that T = 2t and (T — T) 2 = 2t 2 , so that we
can rewrite _
,  ( T  ~T) V*r 2 d\T — T)
2(tt) 3
P(T) dT = e < r *W \ t H ' (136)
186 A Survey of Nuclear Reactions [Pt. VI
Now the probability for the multiple emission is a compounding of nor
mal distribution. As in the familiar statistical problem of the chisquare
«i _
test, we write x 2 = 53 (Ti — T) 2 /4t 2 and recall that x 2 is then the
1
square of the radius vector in a hyperspace of n — 1 dimensions. In
such a space the volume element is proportional to the quantity x" 2 dx
The proportionality constant is of course a function of n. If we recall
that, as the excitation W grows without limit P(n, W) — > 1 for any given
value of n, we can evaluate this function of n. Now for not too small
values of (W — nB)/r and of n, the region of integration, which is
bounded by the coordinate axes and a set of planes in the hyperspace,
can be replaced without serious error of a sphere of radius say Xm in the
polar coordinate system described by x Remembering the required
normalization, we obtain
/Xmax /*'max
X n ~ 2 e~ x d x =f{n)\ t in *>' a e* dt (137)
which we can write P(n,W) = y[(n  l)./2, * max (W)]/r[(ra  l)/2],
where y(n,x) is the incomplete gammafunction defined by
y(n,x) = j e~H n ~ l dt and Y{n) = y(n, #»«>)
It is important to note that we want the dependence of P(n,W) not on
the value of W, as in the usual statistical problem, but on the parameter
n. We can estimate the distribution in x 2 > for any fixed large value of n,
by the familiar saddlepoint method, getting a Gaussian distribution
given by e^* 2 ^ 1 )] 2 /^!) 2 dy ? n ow t jj e fi rs t approximation to P(n)
can be obtained by considering the geometrical interpretation of the
integral, and observing that P(n) is unity for any n such that the sharp
peak in x 2 lies within the original region of integration. Taking the
original limits, then, and using the hyperplane rather than the sphere
boundary, we can estimate that x 2 (H r ,w) = (W — nB)/2r. In the
next approximation, the width of the region in which P(ri) falls from
unity to zero can be estimated from the standard deviation in x 2  We
are led to these results for the mean number of nucleons emitted in the
process and for the standard deviation, with an approximately Gaussian
distribution in n:
W + 2r
ll S
2r + B
(138)
= „ 8t 2 W+2t
n ■ — n ~
B 2 2r + B
Nuclear Reactions at High Energy
187
Sec. 11D]
The relative rms fluctuation is just An/n ~ 3/(n) M  (r/B). The mean
energy loss for the emission of a single Maxwellian particle is B + It,
and the manyparticle case is seen to give a mean value not very different
from what one would get neglecting any correlation, namely W/(2r + B).
The standard deviation is also not much different from the consequence
of a very naive statistical estimate. Some reactions have been observed
in which a highly excited nucleus emits a considerable but definite
a a
O «3
■§ °
PL,
0.3
y —
^* *v
W . t . , = 100 Mev
initial
0.2
_ / /
/ /
/ /
X^V
/ /
N. \
0.1
1
2 4 6
Number of evaporated particles (w)
Fig. 39. The probability of emission of exactly n evaporation particles as a function
of n. These fluctuations are calculated, on a very much simplified model, for a
nucleus of initial A = 100 and excitation energy of 100 Mev. The dashed curve
comes from an attempt to integrate the coupled equations of evaporation; the solid
curve, which is a Poisson distribution, from a still simpler theory. Only qualitative
conclusions may be safely drawn from the present theory. From (Tl) and (H10).
number of identical nucleons, and the energy spectrum for those emitted
particles seems to resemble the statistical energy distribution P(n,W)
given above. 1
The omission of all the complicating effects from this picture is all too
clear. A couple of attempts to solve the cascade equations under more
realistic assumptions have been made (H10, Tl). Even these more
elaborate efforts have neglected the change in mass and in binding energy
as the cooling proceeds, and have in general neglected the fluctuation in
energy taken off with each nucleon. The resulting fluctuations primarily
emphasize the choice the cooling nucleus makes between protons and
neutrons, with differing mean probabilities and mean energy removal
for the two types of nucleons emitted. The results indicate something
very close to a Poisson distribution in the number of neutrons or in the
number of protons emitted, with the mean being given by the mean
cooling behavior we have already considered. In Fig. 39 we plot some
consequences of the two methods mentioned. It is fairly clear that most
1 Private communication from E. Segre.
188 A Survey of Nuclear Reactions [Pt. VI
of the details of the process have been suppressed in these generalized
models. Whether the fluctuations are underestimated or overestimated
by these models is not clear. The changes in mass number and energy
which have been neglected certainly cause correlations to be more im
portant, but some of the correlations have the effect of stabilizing the
process closer to the mean. It is very likely here also that a Monte
Carlo procedure will be the first to yield a reliable answer to the problem.
E. Mesons : Virtual and Real
We have come to the end of a sufficiently long, but yet only very sum
mary, account of the theory of nuclear reactions in general. It is ap
propriate to close with an admonition.
All our considerations have been based on a single picture of the
nucleus: physically, as a collection of neutrons and protons with more or
less strong interaction; formally, as a system whose Hamiltonian was a
sum of kinetic energies of a given number of heavy nucleons plus more
or less complicated interaction potential energies. It is clear that this
idea, taken either physically or formally, is at best an incomplete model.
Looking at the nuclear system in short time intervals, or with fine
distance resolution, we would not in fact distinguish an unchanging
number of nucleons colliding and shifting position, but fixed in number
and type. On the contrary, we know that even the exchange of mo
mentum which is the ordinary force between nucleons, like the exchange
of spin and of charge which correspond to more subtle but still phe
nomologically described interactions is not to be understood in this way.
The mediation of some kind of field of mesons seems certain. Like all
fields whose quanta need relativistic description, that of the mesons
within the nucleus cannot be assigned a definite particle count. We have
to think of transient states, in which mesons appear and disappear,
carrying momentum, charge, and spin back and forth among the heavy
nucleons in what may be a very complicated manner. In a way we have
used a description of the nucleus whose molecular analogue would be a
pair of protons and an oxygen nucleus held together by some given
potential, to form a molecule of water. Yet the meson field is still more
complicated than that of the atomic electrons, though it contains charges.
It is in some ways more closely akin to the essentially relativistic electro
magnetic field, in the cases where the photons may have a purely tran
sient existence, as ordinary photons do in fact have within the nearzone
field of a radiating dipole. The anomalous nucleon magnetic moments
arise from such meson fields. The existence of manybody forces, of ve
locitydependent forces, of circulating currents not ascribable simply to
orbital motion or to spin nipping of the nucleons is the very least to be
Sec. HE] Nuclear Reactions at High Energy 189
expected from the virtual mesons within the nucleus (SI). Suspected
are severe changes in the familiar properties of free nucleons themselves
due to their immersion in the virtual meson sea (M8) . Transient changes
in charge — producing doubly charged protons or negative neutrons —
changes in spin, even in rest mass, are all probable circumstances of the
nuclear matter, looked at sharply enough. And, when energy enough
becomes available from outside to satisfy the demands of permanent
meson liberation, all sorts of new effects can be seen. Meson absorption,
emission, and scattering, where the mesons fly free of the nuclear region
to reach our counters or the emulsion grains, are but the most obvious
effects. All have been studied, and are in active development. What of
the release of mesons, strongly interacting with nucleons as they do,
which never leave the nucleus, but are reabsorbed within it to transfer
energy, momentum, spin, and charge throughout the nuclear drop? Al
ready it seems that the sharp rise of the deuteron photocrosssection
(B6) in the energy region near and just above the threshold for free meson
production involves some such explanations.
All this is far from our simple picture of a closed and tight cluster of
Z protons and N neutrons. But it is a picture closer to the truth. It is
well to end this account, then, with the clear warning that application
of the simpler ideas is reliable at best only in the domain where such
long time averages are involved that the transient mesons can usually
be replaced by the smooth forces of our picture, and for energies well
below the energy at which real mesons can be released, even internally.
With that warning not forgotten, the theory of nuclear reactions can
yield results valuable for innumerable applications, for the detailed
understanding of nuclear structure in a kind of chemist's approximation.
One day this theory may serve for the construction of a better and super
seding picture, based on deeper knowledge of the fundamental nature of
the nucleon itself.
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192 A Survey of Nuclear Reactions [Pt. VI
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Pt. VI]
Appendix I
193
APPENDIX I
In this appendix we have collected graphs which permit the collection of such
cross sections as are plotted in text Figs. 12 through 14. These are mainly graphs
auxiliary to the calculation of penetrabilities. Complete definitions and theory are
found in text references (C5, K7, Y5).
5.0
ffiff0.15fm.t_
4.0
Up lpjflfjg
'Tl'(?)'i$$$f
3
HC I 06$1
"V ( rr}
2.0
IHo'B
^Po°7
BoM
M0.9lllHlllf
3
1.0
0.2
0.4
X=E/E barri( . r
0.6
0.8
Fig. Al. Function for computing the penetrability of the Coulomb barrier without
angular momentum, Po, for 1 = 0. The penetrability Po is defined as: Pq = e~ 2c °
with Co = <7y(x). y(x) is here plotted. Figures 12d and 13a of the text give barrier
heights for various nuclei, g is plotted in Figs. A2a and A2b. After (B13).
194
A Survey of Nuclear Reactions
[Pt. VI
03
.a
a
03
Pt. VI]
Appendix I
195
03
a,
"3
J3
Ml
196
A Survey of Nuclear Reactions
[Pt. VI
to
I
1.0
1 1
X=Vh
Vi
V2 <
1
1.0
1.5
o.c
s = 2.0j
1.0
y=l(l + l)/g*
2.0
3.0
Fig.
A3. The
2ClXx.v)
penetrabilities for
angular momenta I ^ are defined as
Pi = e _zo " x ' ! ". The plot gives (Ci — Co)/?, where g is the characteristic orbital
momentum plotted in Figs. A2a and A2b as a function of two parameters: energy
in terms of the barrier height x and parameter y = 1(1 + l)/<7 2 . After (K7).
Ok
20
&
**
s*%
&
X
"1
€
15
11
20 30 40 50 60 70 80 90
Z
X
sAU
t w
c\e
6
rvW^
»;
V,
, F
'~*2£i
•pvojjii"
Pr
ot<
)I\E
*=0.15_
.20
30
40
50
60
70
80
90
Kg. A4. Values of orbital angular momentum l c , such that the contribution to
the total cross section for compound nucleus formation, a = ttX 2 2 (21 + l)Pi,
10
for all I greater than l c , is less than 10 percent. These curves form a guide for stopping
the computation of the Pi as I grows.
197
198
z
(atomic
number)
100
90
80
70
60
50
30—
25
20—
A Survey of Nuclear Reactions
[Pt. VI
10—
9
8
7
6—
5
(a)
E
in Mev
(center of.
mass)
30
20—
15
10
9
8
7
6
5
2
1.5
1.0
.9
.8
.7
.6
.5
.4—
.1—
X
V
and
d
1.0
.7
.6
.5
1.0
— .7
— .6
(a')
.1—
.09
 .2
(a")
10"
10"
'2 10"
■I io«
5
2 10" s
4
6
8 io 2
2
4
6
8 .i
.2
.3
Hb)
.5
.6
 .7
 .9
 .95
a p
.2
.09—
.1
.15
.2—
.8
— .4
.09
.10
.15
.2
 .7
(c)
.95
Fig. Ao. Nomogram constructed for the evaluation of the penetrability of the Coulomb barrier, in the
case of zero orbital angular momentum, I = 0. The penetrability Po can be found in two ways, of
different accuracy: (1) Using lines (a), (a'), and (a"), the value of the parameter x, which is the energy
of the particle measured in terms of the barrier height for a definite Z, can be found directly. The
approximation A = 2Z is used for this graph, which is quite rough for the heavy elements, and may lead
to Po in error by a factor of ~S, even where Po ~ 0.1. With this value of x, connecting points on lines
(a), (6), and (c) then gives the penetrability, P . (2) The barrier height B may be found, not from the
nomogram, but by directly reading off its value from the graphs of Figs. 12c and 13c of the text. Then
x = E/E, can be found, and connecting proper points on (a), (6), and (c) of the nomogram will give Po
to good accuracy over the whole readable range of the nomogram.
Pt. VI] Appendix II 199
APPENDIX II
We list first a number of theoretical and experimental survey papers of both a
general and a more specialized kind.
Compilations of Experimental Information
M. S. Livingston and H. A. Bethe, Revs.. Modern Phys., 9, 245 (1937). List and
discussion of all work on reactions up to July, 1937.
J. Mattauch and S. Fliigge, Nuclear Physics Tables, New York, Interscience, 1946.
Some data and references for all reactions up to about 1941.
W. Hornyak, T. Lauritsen, P. Morrison, and W. A. Fowler, Revs. Modern Phys.,
22, 291 (1951). A very full survey up to the middle of 1950 dealing with reac
tions involving the lightest target nuclei only, from H to Ne.
Books
S. Devons, The Excited Slates of Nuclei, Cambridge University Press, 1949. Excel
lent account of experimental methods and of theory.
V. F. Weisskopf, in Lecture Series in Nuclear Physics, MDDC1175, U. S. Govern
ment Printing Office, 1947, pp. 106 et seq. Theory.
J. Blatt and V. Weisskopf, Theoretical Nuclear Physics, John Wiley & Sons, New
York, 1952. Comprehensive account.
Papers of More Restricted Scope
(a) Photonuclear Reactions
Experimental
K. Strauch, Phys. Rev., 81, 973 (1951). Work with ~300Mev y's.
A. K. Mann and J. Halpern, Phys. Rev., 82, 733 (1951).
L. Katz and A. S. Penfold, Phys. Rev., 81, 815 (1951).
Theoretical
J. S. Levinger and H. A. Bethe, Phys. Rev., 78, 115 (1950).
E. P. Courant, Phys. Rev., 82, 703 (1951).
(b) Theory of Particle Reactions in General
V. F. Weisskopf, Helv. Phys. Acta, 23, 187 (1950). Physical picture.
E. P. Wigner, Am. J. Phys., 17, 99 (1949). Introduction to the general dis
persion theory.
E. P. Wigner and L. Eisenbud, Phys. Rev., 72, 29 (1947). Presents the full
theory.
J. Jackson, Phys. Rev., 83, 301 (1951). A simplification of the theory.
T. Teichmann, Phys. Rev., 77, 506 (1950).
E. Wigner, Phys. Rev., 73, 1002 (1948).
Thelast two present instructive and useful applications of Wigner s gen
eral theory.
(c) Neutron Excitation
Theoretical
E. Fermi and L. Marshall, Phys. Rev., 71, 666 (1947). Thermal neu
trons; includes experimental work.
B. T. Feld, Phys. Rev., 76, 1115 (1949). Fast neutrons.
H. Feshbach, D. Peaslee, and V. Weisskopf, Phys. Rev., 71, 145, 564
(1947). The basis for our version of the theory of dispersion.
200 A Survey of Nuclear Reactions [Pt. VI
Experimental
R. K. Adair, Revs. Modern Phys., 22, 249 (1950). A comprehensive
review.
D. J. Hughes, W. Spatz, and N. Goldstein, Phys. Rev., 75, 1781 (1949).
(d) Charged Particle Excitation
Theoretical
V. Weisskopf and D. Ewing, Phys. Rev., 57, 472, 935 (1940).
D. Peaslee, Phys. Rev., 74, 1001 (1948). Deuterons only.
Experimental
E. L. Kelly, thesis, University of California, Berkeley, UCRL1044, 1951.
S. N. Ghoshal, Phys. Rev., 80, 939 (1950).
P. C. Gugelot, Phys. Rev., 81, 51 (1951).
(e) Nuclear Fission
Theoretical
N. Bohr and J. Wheeler, Phys. Rev., 56, 426 (1939). General theory,
using drop model.
D. Brunton, Phys. Rev., 76, 1798 (1949).
Experimental
Plutonium Project, Revs. Modern Phys., 18, 513 (1946).
J. Jungerman and S. C. Wright, Phys. Rev., 76, 111 (1949).
J. Jungerman, Phys. Rev., 79, 632 (1950). Alpha and alphainduced
fission.
(f) HighEnergy Nuclear Stars
Theoretical
W. Horning and L. Baumhoff, Phys. Rev., 75, 370 (1949).
M. L. Goldberger, Phys. Rev., 74, 1269 (1948).
Y. Fujimoto and Y. Yamaguchi, Prog. Theor. Phys., 4, 468 (1950).
Experimental
G. Bernardini, E. Booth, and S. Lindenbaum, Phys. Rev., 80, 905 (1950).
400Mev protons.
R. Menon, H. Muirhead, and O. Rochat, Phil. Mag., 41, 583 (1950). Pi
mesoninduced' stars.
Papebs on Specific Reactions
The reactions are grouped by their means of excitation (list closed July, 1951).
The references for each reaction are coded according to the first two or three letters
in their authors' names and the years the works were published. A star (*) means
that work above 100 Mev is reported. Some conventions are worth pointing out:
1. Strictly speaking, gammarays often are secondary products of other reactions.
These are not separately considered. Only reactions in which gammas are the sole
product are listed under "Gammarays."
2. Electroninduced reactions in which the electron is in fact captured by inverse
betadecay have not yet been observed. The electron serves to excite the nucleus,
and itself passes on. It comes away from the reaction, though it is not in fact a
reaction product.
3. Reactions with more than two product particles are listed only once; the order
of emission (if known) is not taken into account.
The classification here adopted depends upon the facts: (i) for evident reasons of
effectiveness and availability, the incident particles used in almost all nuclear reac
Pt. VI] Appendix II 201
tions are light nuclei, up to helium (plus photons and electrons, etc.); and (ii) most
reactions yield for at least one of the products another light nuclear particle, leaving
behind as the second product particle a residual nucleus not far removed from the
target nucleus in charge or mass number.
Gammarays
( T , T ) Sc46; Wi45b; Wi45c; Gu41; Gae49.
( T , meson) *Lax51; *Lit51; *Mo50; *Pet51; *Ste50.
( T , n) Bal46; *Per49; Mc49; By51; Jo50; Kat51b; Med50; Pri50.
(7, 2«) Bal46; *Per48.
( T , p) Same as (7, n) Hi47; Cou51; Di50; *Lev51; To51; *Wak51; By51.
( T ; p, n, or d) Bal46; *Per48; By51.
(7, a) Has51; Mil50; Pr50.
(7, multiple products) Bal46; *Per48.
(7, star) *Gae50.
(7, fission) Ti49; Hax41; Bo39; Ko50; *Su50.
Electrons
(e; e', n) B149; Sk48.
(e; e') Wi44; Wi45a; Wi45c; Mul51.
Mesons
O, it) *Ca51; *Be50.
(*, star) *Be50; *Che50; *Men50; *Ta50.
Neutrons
(n, 7) Fes47; *Kn49; Hug49; Gos47; An50; Cap51; Ham50; He50; Hu51;
Mu50; Ki51.
(n, n) Fe47; Fel49; Se49; Fes47; Ad49; Har50; *Pas50; Stt51.
(n, In) Hou46; Ma42; Hey37; Coh51; Fow50; Waf50.
(n, spallation) De48.
(n, p) Coo49; Boo37; Am35; Sa40; Coh51.
(n; p, n, or d) *Kn49; *Bru49; *Chw50; *Ha50; Waf50.
(n; H 3 or He 3 ) Bru49; Cor41; On40; *Ha50.
(n, a) Sh41; Wu40; *Kn49; Am35; Sa40; B046; St50.
(n, stars) *Go48; *Tr50.
(n, fission) P146; *Ke48; Ph49; Fra47; Bou50; Ros50.
Protons
(p, 7) Du38; Wal48; Ben46; Fo48; Dev49; Hal50.
(p, t) *Bj50; *Blo51; *Hen51; *Jon50.
(p, n) Wei40; Du38; Del39; Bla51; *Bod51; *Kn51; *Mir51.
(p, In) Te47a.
(p, spallation) *Baz50; *Baz51; *Hy50; *Me51.
(p, p) Fu48; Ba39; Hei47; Bed49; Goh51; Lth50; Rh50.
(p; p, n, or d) Ri46; Pa48; Th49.
(p, a) Th49; Bur49; Bar50; Ch50; Coc49; Ra50; Dev49.
(p, heavy particle) *Mar51; *Wr50.
(p, star) *Tho49; *Cam50; *Fr50; *Hod51; *Pek50.
(p, fission) Ju48; Bo39.
Deuterons
(d, 7) We43.
(d, n) Pe48; Ke49; C146b; Ro47; Amm49; Fa49; *Kn51.
(d, 2ra) Ke49; Te47a.
202
A Survey of Nuclear Reactions
[Pt. VI
(d, spallation) Wil48; *Mi48; *Bat50; *Lin50.
(d, p) Pe48; Pol49; C146b; A1148; Buw50; Cu50; Hav51; Phi50; Va51.
(d, pa) C147.
(d, d) Gug47; Gr49; Ker51.
(d, H 3 ) Kr41; Wi46; Ka49.
(d, a ) Kri49; Li38; C146a; In50; Sch50.
(d, an) Mad50.
{d, stars) *Ga49a; *Ho49.
(d, fission) *Wo49; Ju48; Kr40; *Goe49.
Tritium
(H 3 , n) Cr51.
(H 3 , p) Ku48a.
(H 3 , He 3 ) Ku47b.
He 3
(He 3 , p) A139.
Alphaparticles
(a, 7) Ben51.
(a, mesons) *Bu49; *Jon51.
(a, n) Eg48; Bra47; Rid37; Rie48; Hap49; Tem49.
(a, 2n) Gh48; Te47a; Fi50; Tem49.
(a, spallation) *Wo49; *Oc48; *Lin50.
(a, p) Bro49; Roy51.
(or, a) La39.
(a; p, 3n) Hel46; Ne49; Te47b.
(a, stars) *Ga49b.
(a, fission) *Oc48; *Wo49; Ju48.
Heavy particles
(heavy particle, star) *Bra48; *Bra49.
REFERENCES FOR APPENDIX II
(Ad49) R. K. Adair, C. K. Bockelman, and R. E. Peterson, Phys. Rev., 76, 308
(L) (1949).
(A139) L. W. Alvarez and R. Cornog, Phys. Rev., 56, 613 (L) (1939). (Si)
(A1148) H. R. Allan and C. A. Wilkinson, Proc. Roy. Soc. (London), A194, 131
(1948).
(Am35) E. Amaldi, O. D'Agostino, E. Fermi, B. Pontecorvo, F. Rasetti, E. Segre,
Proc. Roy. Soc. (London), A149, 522 (1935).
(Amm49) P. Ammiraju, Phys. Rev., 76, 1421 (L) (1949).
(An50) H. L. Anderson, Phys. Rev., 80, 499 (1950).
(Ba39) S. Barnes and P. Aradine, Phys. Rev., 55 (1950). (In)
(Bal46) G. Baldwin and G. Klaiber, Phys. Rev., 70, 259 (1946). (Excitation
survey)
(Bar50) C. A. Barres, A. P. French, and S. Devons, Nature, 166, 145 (L) (1950).
(Bat50) F. O. Bartell, A. C. Helmholz, S. D. Softky, and D. B. Stewart, Phys.
Rev., 80, 1006 (1950).
(Baz50) R. E. Batzel and G. T. Seaborg, Phys. Rev., 79, 528 (L) (1950).
(Baz51) R. E. Batzel and G. T. Seaborg, Phys. Rev., 82, 607 (1951).
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Pt. VI]
Appendix II
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204
A Survey of Nuclear Reactions
[Pt. VI
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Pt. VI]
Appendix II
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206
A Survey of Nuclear Reactions
[Pt. VI
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Pt. VI]
Appendix II
207
(Se49) F. G. P. Seidl, Phys. Rev., 75, 1508 (1949).
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PART VII
The Neutron
BERNARD T. FELD
The Massachusetts Institute of Technology
In 1948, when this work was begun, the status of the field of neutron
physics was uncertain and rather anomalous. Important progress,
which had been made during the war, was known to the great body of
physicists only through a few "releases" and through the Smyth report.
The main prewar references — the articles of Bethe, Bacher, and Liv
ingston, in Reviews of Modern Physics — were hopelessly outofdate.
They had been partially and inadequately replaced by hastily assembled
and informally distributed mimeographed notes of a Los Alamos lecture
series on nuclear physics (LA24) and of a series of lectures by Fermi on
neutron physics.
By the time the first draft of this work was completed, in June of
1949, the situation was quite a different one. As a consequence of a
wise and farsighted policy on the part of the American, British, and
Canadian atomic energy commissions, practically all the basic scientific
data, which had been accumulated during the war, appeared in the open
literature. As a result of the widespread renewal of interest in the prob
lems of neutron physics, the field has developed, and continues to de
velop, at a rate which has converted the task of compiler and author
into an almost hopeless struggle against obsolescence and has expanded
the dimensions of this work far beyond its original conception. The
unequal struggle was, quite arbitrarily, concluded as of July 1, 1951.
Since then, some changes have been made, mainly as a result of impor
tant new developments which have come to the attention of the author,
mostly through publication in American journals.
This work is primarily intended for the practicing nuclear physicist.
It presupposes a knowledge of the fundamentals of nuclear physics as
well as of its terminology. Some of the terms, such as "barn," 1 are of
comparatively recent origin, but their use has become widespread.
1 Barn, a unit of cross section; origin: big as a . (1 barn = 10 24 cm 2 .)
This unit is said to have been invented by the nuclear physicists at Purdue Uni
versity, around 1941 or 1942, to describe nuclear cross sections which are relatively
easy to measure.
208
Sec. 1A] Properties and Fundamental Interactions 209
Although an attempt has been made to present herein a complete
summary of the major aspects of neutron physics, some important
applications have, perforce, received inadequate treatment or been
omitted completely. Thus, no attention has been paid to the role of
neutrons in biophysics; nor has any space been devoted to the many
interesting problems concerning the effects of neutron interactions on
the macroscopic physical properties of matter. Also, and regretfully,
omitted is a discussion of the fascinating problems of the origin of the
elements, in which the properties of neutron interactions may have
played a decisive role. [The interested reader is referred to an excellent
summary of this field by Alpher and Herman, Revs. Mod. Phys., 22, 153
(1950).]
It would be impossible to give a complete list of the many individuals
to whom I am indebted for aid, encouragement, information, discussion,
and criticism during the progress of this work. My colleagues at M.I.T.,
Brookhaven, and elsewhere, especially H. Feshbach, M. Goldhaber,
G. Placzek, and V. F. Weisskopf , have been unusually generous in this
respect. I owe a special debt to E. Amaldi, of the University of Rome,
not only for congenial and informative discussions but for having gen
erously allowed me access to an unpublished work on neutron physics
by G. C. Wick and himself.
In memory of many pleasant hours, during which the outline of this
article achieved shape and substance, I respectfully dedicate this work
to my teacher and friend, the late H. H. Goldsmith.
SECTION 1. PROPERTIES AND FUNDAMENTAL
INTERACTIONS
A. Discovery
The discovery of the neutron is one of the most dramatic chapters in
the history of modern physics. It started in Germany, in the last month
of 1930, with the report by Bothe and Becker (B50) of a penetrating
radiation resulting from the bombardment of certain light elements by
polonium alphaparticles. They bombarded many substances, using a
Geiger pointcounter as a detector of the resulting radiation. Most of
the elements investigated (Pb, Ag, Ca, N, C, O, Ne) yielded no detect
able radiation; two (Mg, Al) showed a slight effect. Lithium, boron,
and fluorine gave appreciable amounts of radiation capable of affecting
the counter, and beryllium yielded a comparatively tremendous amount.
Bothe and Becker concluded that the radiation consisted of gamma
210 The Neutron [Pt. VII
rays, more penetrating than any that had been observed up to that
time. 1
Curie and Joliot (C36, J10) immediately undertook a study of the
properties of this penetrating radiation. They had available a much
stronger polonium source (100 millicuries, as compared to the 37
available to Bothe and Becker) and were able to measure the absorp
tion of the radiation in lead. They observed, for the radiation from
beryllium, an exponential attenuation with an absorption coefficient of
0.15 cm 1 . (The radiations from boron and lithium had lead absorp
tion coefficients of 0.2 and 1.7 cm 1 , respectively.)
So far, there was nothing to contradict the suggestion — and, indeed,
this was universally assumed — that the radiations were very penetrat
ing gammarays. (We know now that an absorption coefficient in lead
of 0.15 cm 1 is smaller than that of the most penetrating gammarays.)
However, in January of 1932, Curie and Joliot (C37) reported the fol
lowing interesting observations : They investigated the effects of placing
thin screens of various materials in front of the ionization chamber,
which was being used to detect the radiations. For most of the screens
nothing noteworthy occurred. However, when the screens contained
hydrogen, the current in the ionization chamber went up.
Curie and Joliot inferred that the increased ionization was due to the
ejection of protons from the screen by the primary radiation. They
strengthened this theory by the following set of observations: (1) The
application of a magnetic field in the region between the screen and the
ionization chamber did not decrease the effect; it would have, if the
ejected particles had been slow electrons. (2) The effect vanished when
0.2 mm of aluminum was placed between the screen and the chamber.
This was sufficient to absorb protons, but not fast electrons. (3) Cloud
chamber photographs of the tracks of particles ejected by the radiation
1 The conclusions of Bothe and Becker, as well as the observations upon which
they were based, turn out, in retrospect, to be completely valid. The Geiger point
counter, used as a detector by these investigators, was not sensitive to neutrons;
thus, Bothe and Becker could not have observed neutrons in their experiments. It
was only with the introduction of ionization chambers, proportional counters, and
cloud chambers as detectors that the neutrons became observable in subsequent
investigations. The measurements of the gammaray energies (~3 Mev from boron
and ~5 Mev from beryllium), reported by Bothe and Becker, have subsequently
been confirmed. Indeed, the important discovery by Bothe and Becker of artificial
excitation of nuclear gammaradiation, reported in their 1930 paper, has unfortu
nately been almost completely obscured by the drama associated with the discovery
of the neutron. For a more complete discussion of the significance of the observa
tions of Bothe and coworkers, the reader is referred to an article by Fleischmann
(F32).
Sec. 1A] Properties and Fundamental Interactions 211
from hydrogencontaining screens showed ionization consistent with
that of protons, but inconsistent with that of electrons (C38).
Curie and Joliot first hypothesized that the ejection of protons from
the screens was due to Compton scattering of the incident gammaradia
tion by the hydrogen nuclei. From the observed range (energy) of the
(recoil) protons, they estimated the energy of the photons from beryl
lium to be 50 Mev. Curie and Joliot recognized a number of serious
difficulties in connection with their hypothesis; they could conceive of
no source of such highenergy photons in a reaction of alphaparticles on
beryllium, and they felt (erroneously) that photons of such high energy
should be even more penetrating than the observed radiation. But,
while they decided that the ejection of protons (and other light nuclei)
was by a new type of gammaray interaction (C38), it remained for
Chadwick, working at the Cavendish Laboratory in England, to reject
the gammaray hypothesis and take the bold step of postulating a new
particle.
Chadwick was very quick to follow up the researches reported from
the Institut du Radium. Only slightly more than a month after the
report discussed above he was publishing data (C5), obtained with
counters and cloud chambers, showing that the radiations from beryl
lium bombarded with alphaparticles were capable of conferring high
speeds not only upon protons, but also upon the nuclei of other light
elements (He, Li, Be, B, C, N, 0, A). From the observed ranges of the
light nuclei, and using the then current rangeenergy relationships,
Chadwick showed that the Compton recoil hypothesis of Curie and
Joliot was inconsistent with the data. The data could, however, be
explained if the light nuclei were assumed to be recoils from elastic
collisions with a neutral (to explain the great penetrability) particle
of approximately protonic mass.
To obtain the mass of this particle (called by Chadwick the neutron)
Chadwick (C6) used the available data on the maximum range (velocity)
of the proton recoils, and the results of Feather (F2) on the maximum
range of the nitrogen recoils, observed in a cloud chamber. (If these
were due to Compton recoils, they would have required gammaray
energies of 55 and 90 Mev, respectively.) By application of the law of
conservation of momentum, Chadwick derived that the particles re
sponsible for these recoils had a mass of 1.15 times the proton mass, with
an uncertainty such that "it is legitimate to conclude that the mass of
the neutron is very nearly the same as the mass of the proton."
Another estimate of the mass of the neutron was made by Chadwick
from observations on the neutronproducing reaction B 11 + He 4 — > N 14
+ n 1 . On the assumption that the maximumenergy neutrons (obtained
212 The Neutron [Pt. VII
from the maximum range of proton recoils) correspond to leaving N 14 in
its ground state, and from the values of the masses of the three nuclei
involved, previously determined by Aston, Chadwick deduced the value
of 1.0067 atomic mass units for the mass of the neutron.
Within a short time, immediately following upon the series of an
nouncements described above, a large number of investigators in many
lands were conducting experiments on the properties of the neutron and
its interactions. Although it is not the author's intention to continue
this historical survey much beyond the discovery stage, it is of some
interest to review briefly the advances made within only a year after
the appearance of Chadwick's papers. For a more detailed discussion
of the early history, the reader should refer to the original papers and
to excellent review articles (written in 1933) by Chadwick (C7) and
Darrow (D4).
It was soon ascertained that the neutrons resulting from the bom
bardment of beryllium and boron by polonium alphaparticles had a
rather wide energy spread (F2, M21, C39) including many neutrons of
energy considerably below 1 Mev (A36), with the neutron energy
strongly dependent on the direction of emission, with respect to the
alphaparticle direction, as well as on the energy of the alphaparticle.
Gammarays were also shown to be emitted in the neutronproducing
reaction (B51, C39).
The yield of neutrons from beryllium and boron was found to decrease
rapidly with decreasing alphaparticle energy (R7, C40, C7).
By placing considerable quantities of lead next to the neutrondetect
ing ionization chamber (but not directly in the beam) (B70) or by sur
rounding the cloud chamber with copper (A36), the number of neutrons
detected was appreciably increased, thus indicating a large neutron
scattering by these substances.
In the nitrogen gas of the cloud chamber, used in the experiments of
Feather (F2), a number of events were observed which could be as
cribed to the transmutation N 14 + n 1 — ► B 11 + He 4 , the reverse of
the reaction on boron which had been observed to produce neutrons.
He also observed transmutations in oxygen and carbon (F3). Transmu
tations were observed in nitrogen, oxygen, and aluminum by Meitner
and Philipp (M21), and in nitrogen by Harkins, Gans, and Newson
(H37) and by Kurie (K27).
At first, it was generally assumed that the neutron is probably a
closely bound combination of a proton and an electron, 1 especially since
1 As early as 1920, Rutherford (R27) published an interesting and prophetic specu
lation: "Under some conditions ... it may be possible for an electron to combine
much more closely with the H nucleus [than in the neutral hydrogen atom], forming
Sec. IB] Properties and Fundamental Interactions 213
the first estimates indicated that its mass is less than that of the proton.
The first suggestion that the neutron should be regarded as a fundamen
tal particle appears to have come from Iwanenko (13), and it soon
became clear, mainly on the basis of quantummechanical arguments
involving the spin and statistics of light nuclei, that the neutron, like
the proton, probably has a spin of J and obeys FermiDirac statistics
(M21, C7). Thus, with the discovery of the neutron and the recogni
tion of its properties, the currently accepted picture of nuclei, as con
sisting of protons and neutrons, soon emerged.
Since 1933 the development of the field has proceeded with rapidly
increasing intensity. We therefore abandon, at this point, the historical
survey. Instead, we shall summarize and discuss the present knowledge
of the properties of neutrons and their interactions. We shall, however,
in discussing each aspect, attempt to include some of the historical back
ground. It must be emphasized that, although, in the relatively short
time that has elapsed since the discovery of the neutron, tremendous
progress has been made in understanding and utilizing it, there is still
much to be done before the neutron can be said to be completely under
stood. Many important and crucial experiments are in the process of
being performed, while others are still in the future.
B. Properties
Since the neutron is one of the constituents of atomic nuclei, a knowl
edge of its properties . is fundamental for the understanding of nuclei.
In addition, the properties of the neutron determine, to a large extent,
the interactions between neutrons and nuclei, and between neutrons and
conglomerations of nuclei (matter). In this section we summarize
these properties from the point of view of the neutron as a fundamental
particle.
Wherever possible, we shall discuss the experimental evidence on
which the conclusions as to the nature of the neutron are based. How
ever, the understanding of many of these experiments depends on a
detailed knowledge of the interaction of neutrons with matter. In such
cases, the results of the experiments will simply be stated, and the dis
cussion of the experiments reserved for subsequent sections.
a kind of neutral doublet. Such an atom would have very novel properties. Its
external field would be practically zero, except very close to the nucleus, and in
consequence it should be able to move freely through matter. Its presence would
probably be difficult to detect by the spectroscope, and it may be impossible to
contain it in a sealed vessel. On the other hand, it should enter readily the struc
ture of atoms, and may either unite with the nucleus or be disintegrated by its
intense field, resulting possibly in the escape of a charged H atom or an electron
or both."
214 The Neutron [Pt. VII
Before proceeding with the discussion of the neutron, a few points
should be noted in justification of devoting so much attention to the
neutron as compared to other nuclear particles. The neutron is un
matched among atomic and nuclear particles as a tool for the investiga
tion of nuclear properties. The distinguishing feature of neutrons is
the absence of electric charge; thus, the interaction of neutrons with
matter is primarily determined by purely nuclear properties.
In the fifty or so years since its beginnings, nuclear physics has made
large contributions to the development of the other sciences (chemistry,
geology, biology, medicine) and to technology. Since its discovery the
neutron has assumed a most important role in furthering this progress.
Were there no other reason, the importance of the neutron for the nu
clear chain reaction might constitute sufficient justification for assign
ing to it a position of special significance among the tools of modern
science.
1. Charge. The neutron is usually assumed to have no net charge.
This assumption is consistent with all the observed properties and inter
actions of neutrons. However, the observations do not preclude the
possibility that the neutron may have a net charge so small as to have
heretofore eluded detection. It is therefore of interest to derive from
the available evidence an upper limit for the magnitude of the possible
neutron charge.
The most direct evidence on the neutron's neutrality comes from the
experiments of Dee (D6), reported at the same time as Chadwick's
announcement of the discovery of the neutron. He investigated the
ionization produced in air in a cloud chamber irradiated by fast neu
trons and concluded that, if the neutron interacts with atomic electrons
at all, this process produces not more than one ion pair per 3 meters of
the neutron's path in air. From these data it may be concluded (F36)
that the charge of the neutron is less than J^oo of the proton charge.
A somewhat less direct determination of an upper limit to the neutron
charge may be obtained from considerations involving the observed neu
trality of atoms throughout the periodic table. Since the ratio of neu
trons to protons in atomic nuclei increases from zero in hydrogen to 1.6
in uranium, this neutrality implies both that the difference between the
proton and electron charges is small and that the neutron charge is
small. An estimate of the, possible magnitude of the neutron charge
depends on the accuracy with which we know atoms to be neutral.
Rabi J and coworkers have observed that the molecule Csl (108 pro
tons and electrons and 152 neutrons) has a net charge less than 10 — 10 of
1 Private communication.
Sec. IB] Properties and Fundamental Interactions 215
the electron charge. From this observation it is possible to draw a num
ber of conclusions: (1) The smallness of the charge of Csl may be due to
an accidental cancellation of the neutron charge and the protonelectron
charge difference. In this case, which is exceedingly unlikely, the mag
nitude of these charges can only be determined by another observation
on some other atom or molecule, with a different neutronproton ratio.
In any event, both the neutron charge and the protonelectron charge
difference would have to be quite small to account for the neutrality of
atoms, say < 10 — 5 e. (2) Either the neutron charge or the protonelec
tron charge difference is zero (or considerably less than 10~~ 12 electron
charges), in which case the nonzero charge is less than 10 — 12 electron
charges. (3) The neutron charge and the protonelectron charge dif
ference are both finite, and very small. Thus, if they were equal and
opposite, the neutron charge would have to be less than 2 X 10 — 12
electron charges. It seems quite reasonable to conclude, from this evi
dence, that the net charge on the neutron is exceedingly small, probably
less than 10 — 12 electron charges.
Despite the electrically neutral character of the neutron, there is a
very small electromagnetic interaction between neutrons and charged
particles arising out of the magnetic dipole moment of the neutron (to
be discussed). In addition, it should be pointed out that, according to
currently prevailing theories, the neutron should not strictly be regarded
as a fundamental particle, but rather as having a complex structure
involving equal numbers of + and — charged particles (mesons).
Such a structure is certainly required for the understanding of the mag
netic moment of the neutron. The complex nature of the neutron would
lead to a small, purely electrical interaction between neutrons and elec
trons; the existence and magnitude of such a neutronelectron interac
tion is being investigated and is discussed further on.
Two other points are worth noting as having a bearing on the possible
charge of the neutron: (1) The observation of the decay of the neutron
into a proton, an electron, and a neutrino (discussed in Section 1B3)
implies that (a), if the proton and electron charges are equal and oppo
site, the neutron and the neutrino have the same charge, if any; or (b),
if the neutrino is uncharged, the neutron charge is equal to the proton
electron charge difference; or (c) some combination of (a) and (b). In
this connection, de Broglie has pointed out (B69) that hypothesis (a)
would be consistent with the neutrality of atoms if there were both posi
tively and negatively charged neutrinos and, correspondingly, neutrons,
so that heavy nuclei could contain approximately equal numbers of neu
trons of the two charges. (2) The smallness of the neutronelectron
216 The Neutron [Pt. VII
interaction (Section 1C3) implies that the charge on the neutron must
be exceedingly small. In particular, the observation by Fermi and Mar
shall (F23) that the scattering of thermal neutrons by xenon atoms is
spherically symmetrical to better than 1 percent allows us to set an
upper limit on the possible net charge of the neutron of >~10~ 18 electron
charges.
2. Mass. 1 The most accurate determination of the neutron mass is
obtained, indirectly, from observations on nuclear transmutations in
which all the masses and energies, except the mass of the neutron, are
known. Thus, the estimate of Chadwick, quoted above, was obtained
from consideration of the B u (a,n)N u reaction (C7). In the same
paper Chadwick observed that a more direct determination of the neu
tron mass could be obtained if the binding energy of the deuteron, the
nucleus consisting of a neutron and a proton, were known, since the
masses of the proton and deuteron had both been determined by mass
spectroscopic means.
Soon afterward Chadwick and Goldhaber (C8) observed that the
deuteron can be decomposed into a neutron and a proton by the absorp
tion of a gammaray from ThC" (2.614 8 Mev). From the energy of the
resulting proton they obtained a value of 2. 1 Mev for the deuteron bind
ing energy. Until recently the accepted neutron mass was obtained
from accurate measurements of the threshold gammaray energy for
the photodisintegration of the deuteron.
However, quite recently Bell and Elliott (B16) have accurately
measured the energy of the gammarays resulting from the capture of
neutrons, of negligible kinetic energy, by protons (the reverse of the
photodisintegration). Using their value of 2.230 ± 0.007 Mev for the
gammaray energy, and other data, Bainbridge (Volume I, Part 5)
gives the value M = 1.008982 atomic mass units for the neutron mass.
3. Instability. With the first relatively accurate determination of the
neutron mass (C8), which showed that it exceeds the mass of the hydro
gen atom, Chadwick and Goldhaber ventured the prediction that the
neutron should be unstable against betadecay, according to the reaction
neutron — > proton + /3~ + neutrino. The maximum betaray kinetic
energy for the decay of neutrons of negligible kinetic energy is given by
the n — H mass difference, which we take to be equivalent to 782 ± 1
1 That the neutron mass behaves in the conventional fashion with respect to gravi
tation has been directly verified by McReynolds (M19), who measured the "free
fall" of slow neutrons.
Sec. IB] Properties and Fundamental Interactions
217
kev (T16). 1 Assuming that the decay of the neutron follows the empiri
cal laws of betadecay, being an allowed transition with a log ft value of
~3.5, the halflife of the neutron should be ~20 min.
Recent experiments in two laboratories have established that the
neutron does decay, as per the above predictions. In the first ones, per
formed at the Oak Ridge National Laboratories by Snell, Miller, Pleason
ton, and McCord (S43, S44), the neutron decay has been observed
through the simultaneous (coincidence) detection of the betaparticle
200 300 400 BOO
Energy (kev)
Fig. 1. Fermi plot of the negatrons from the decay of neutrons, due to Robson
(R15). The deviations from the allowed shape, below 300 kev, are instrumental.
and proton, resulting from the decay of neutrons in an intense beam
passing through an evacuated chamber containing the particle detec
tors. Control experiments were performed to establish that the coinci
dences occurred only in the presence of the neutron beam, and that the
betaparticles were of roughly the expected energy. These experiments
are consistent with "a halflife in the range 1030 minutes."
Robson (R15) at the Chalk River Atomic Energy Project, in Canada,
has not only observed the neutron decay and measured its halflife but,
in an experiment of exemplary care and ingenuity, has also succeeded in
measuring the spectrum of the decay electrons. He first observed the
positive heavy particles resulting from the decays (in a highly sensitive
mass spectrometer) and established that they are protons. He then
1 Actually, in the decay of a neutron at rest the emission of a ^particle and a
neutrino requires, for the conservation of momentum, that some energy be given to
the proton. However, for the emission of a /3ray of the maximum possible energy,
the kinetic energy of the recoil proton is only 0.43 kev.
218 The Neutron [Pt. VII
succeeded in obtaining coincidences between the protons and their
associated betaparticles, detected in a magnetic lens electron spectrom
eter in which he simultaneously measured the energy of the betaparti
cles. The betaspectrum obtained in this manner is shown in Fig. ] , a
conventional allowedtransition Fermi plot. The end point corresponds
to a maximum betaray kinetic energy of 782 ± 13 kev. From an (ex
perimental plus computational) evaluation of the geometrical efficiency
of proton detection and a measurement of the neutron density in the
beam (only ~10 4 neutrons/cm 3 ), Robson deduced the halflife of 12.8
±2.5 min for the decay of the neutron.
Other experiments have been suggested for the observation of the
decay of the neutron and the measurement of its halflife. Among the
most interesting is the following: A sealed, evacuated, thin walled con
tainer of material with low neutron absorption, say glass, is placed in a
region of high neutron density. After some time has elapsed, the
accumulation of hydrogen gas in the container is measured. The con
tainer acts as a semipermeable membrane, permitting the free entry of
neutrons but preventing the escape of hydrogen formed by neutron
decay in the container. For a neutron density of about 0.5 X 10 10
neutrons/cm 3 (corresponding to a thermal neutron flux of ~10 15
neutrons • cm" 2 • sec 1 , of average velocity ~2 X 10 s cm/sec) a hydrogen
pressure of ~10~ 4 mm of Hg would be developed in the container in
approximately one month ; such an accumulation of hydrogen gas should
be easily observable. The difficulties of this experiment involve the
attainment of sufficiently high neutron fluxes (see Section 3), possible
production of hydrogen through (w,p) reactions in the walls of the con
tainer, and the necessity for complete removal of hydrogen from the
container (walls) prior to irradiation.
4. Spin and Statistics. Like the electron and the proton, the neutron
has a spin of \ and obeys FermiDirac statistics. The evidence for the
spin value of \ is quite conclusive, although it involves a combination
of experimental observations and the theoretical deductions therefrom.
Some of these experiments involve the interaction between neutrons
and protons and the properties of the deuteron; they will be further dis
cussed in Section 1C.
The most important evidences for the value of \ for the spin of the
neutron are: (1) the cross section and energy dependence of the scatter
ing of neutrons by protons, taken in conjunction with the evidence con
cerning the neutronproton force, derived from the binding energy and
spin of the deuteron; (2) the values of the magnetic moments of the
Sec. IB] Properties and Fundamental Interactions 219
deuteron, proton, and neutron; (3) the coherent scattering of neutrons
by hydrogen, as evidenced in experiments involving the scattering by
hydrogencontaining crystals, total reflection from hydrogen containing
"mirrors," and the scattering by ortho and parahydrogen; (4) the
polarization of neutrons by scattering in ferromagnetic materials and,
in particular, by total reflection from magnetic mirrors. The results of
these experiments, coupled with the success of the semiempirical theory
of nuclear forces in explaining the observed properties of the deuteron
and many of the properties of light nuclei, constitute a most convincing
argument for the correctness of the assignment to the neutron of a
spin of J.
As a particle of odd halfintegral spin, the neutron is expected to
obey FermiDirac statistics. All the available evidence supports this
expectation. The most important arguments concerning the statistics
obeyed by the neutron involve (1) properties of the light nuclei — in
particular, the fact that the deuteron and alphaparticle are known to
obey BoseEinstein statistics — which can be qualitatively understood
only by use of the exclusion principle (in a fashion closely analogous to
its application to the qualitative understanding of the periodic table),
and (2) the saturation of nuclear forces, as manifested in the constant
density of nuclear matter (the proportionality of the nuclear radius
to A*).
5. Magnetic Moment. 1 The first strong indication that the neutron
has an intrinsic magnetic dipole moment came from the observation
that the moments of the proton and deuteron are very different : nd < n P
by <~2 nuclear magnetons. Since the deuteron is known to have spin
I = 1, it is expected, on the basis of the simplest reasonable deuteron
structure (a neutron and a proton in a 3 Si state), that m = y. p + \x n .
1 The neutron, having spin \, cannot show any moments higher than a dipole
moment. Although the possibility of the existence of an electric dipole moment of
the neutron is usually dismissed, on the basis of theoretical arguments involving the
concepts of symmetry and parity, it has recently been pointed out by Purcell and
Ramsey (P32) that the assumptions on which these arguments are based are still not
completely proved. Thus, for example, "if the nucleon should spend part of its time
asymmetrically dissociated into opposite magnetic poles of the type that Dirac has
shown to be theoretically possible, a circulation of these magnetic poles could give
rise to an electric dipole moment." Although the detection of a possible electric
dipole moment of the neutron is experimentally rather difficult, the above authors
suggested that it could be observed in a modification of the experiment of Alvarez
and Bloch (A13); they proposed to detect a shift of the neutron precession frequency,
caused by the application of a strong electric field. However, such an experiment
by Purcell, Ramsey, and Smith (S40) yielded a negative result and placed an upper
limit, on the neutron's electric dipole moment, of two opposite electron charges sepa
rated by a distance of 510 21 cm.
220 The Neutron [Pt. VII
The magnetic moment of the neutron has been measured by an in
genious modification of the Rabitype molecular beam magnetic reso
nance experiment (Rl). In this experiment the usual polarizing and
analyzing (focusing) inhomogeneous magnetic fields are replaced by
slabs of magnetized iron whose property of preferentially transmitting
neutrons of one direction of polarization is used in a manner completely
analogous to an optical polarimeter. (The polarization of neutrons
by transmission through magnetized iron, the Bloch effect, will be dis
cussed in Section 5D.)
Early measurements of the neutron magnetic moment (F47, P28)
gave a value of <~ — 2 nuclear magnetons. The negative sign, which was
definitely established by observing the direction of precession of neutrons
in a known magnetic field, means that the neutron angular momentum
and its magnetic moment are oppositely directed.
The first accurate measurement of the neutron moment, by Alvarez
and Bloch (A13), yielded the value  n n \ = 1.935 ± 0.02 nuclear magne
tons. More recent experiments (A34, B39, R16) have measured, with
high precision, the ratio of the neutron moment to the proton moment.
These measurements, when combined with the latest value of the pro
ton moment, yield (Ml)
fi n = 1.91280 ± 0.00009 nuclear magnetons
6. Wave Properties. In accordance with the laws of quantum me
chanics we expect that the neutron should, under the appropriate experi
mental conditions, exhibit wave properties. Associated with neutrons
of kinetic energy E (velocity v, momentum p) there is a wavelength
x = 1 = Wev> (for " <<c) (1)
Table 1 gives the neutron wavelengths corresponding to an assortment
of neutron energies. The energy is expressed both in electron volts and
in degrees Kelvin, the temperature T corresponding to a given energy
E being defined by the relationship
E = kT (2)
where E is expressed in ergs and k is Boltzmann's constant, k = 1.3803
X 10 16 erg/°K. Also included in Table 1 is the "Dirac wavelength,"
X = X/2t, which is frequently the form in which the neutron wave
length enters into the expressions with which we shall be concerned.
The classification of neutrons into types, according to their energy, will
be described in Section 1B7.
Sec. IB] Properties and Fundamental Interactions
TABLE 1
Wavelengths Associated with Various Neutron Energies
221
E(ev)
T(°K)
v (cm/sec)
X (cm)
X (cm)
Type
0.001
11.6
4.37 X 10 4
9.04 X 10" 8
1.44 X 10" 8
Cold
0.025
290
2.19 X 10 5
1.81 X 10" 8
2.88 X 10" 9
Thermal
1.0
1.16 X 10 4
1.38 X 10 6
2.86 X 10 ~ 9
4.55 X 10 10
Slow (reso
nance)
100
1.16 X 10 6
1.38 X 10 7
2.86 X 10" 10
4.55 X 10 11
Slow
10 4
1.16 X 10 8
1.38 X 10 8
2.86 X 10" 11
4.55 X 10" 12
Intermediate
10 6
1.16 X 10 10
1.38 X 10 9
2.86 X 10 12
4.55 X 10 " 13
Fast
10 8
1.16 X 10 12
1.28 X 10 10
2.79 X 10~ 13
4.43 X 10 14
Ultrafast
10 10
1.16 X 10 14
2.99 X 10 10
1.14 X 10 14
1.81 X 10~ 16
Ultrafast
(relativis
tic)
For the purpose of rapid calculation it is convenient to note the fol
lowing relationships:
r(in °K) = 1.16 X 10 4 #
Kin cm/sec) = 1.38 X 10 G E y °
X(in cm) = 2.86 X lO 9 ^ K
X(in cm) = 4.55 X 10 10 £ H
where E is always given in electron volts:
1 ev = 1.602 X 10"
ergs
These expressions hold only in the nonrelativistic energy region, i.e.,
for kinetic energies well below the energy corresponding to the neutron
rest mass,
Mc 2 = 939.5 million electron volts (Mev)
As can be seen from Table 1, the relativistic effects (on the connection
between v, or X, and E) are already evident, albeit still small, at a kinetic
energy of 100 Mev.
In general, the importance of the wave characteristics of the neutron
is determined by the magnitude of the ratio between the neutron (Dirac)
wavelength and the dimensions of the system with which the neutron
is interacting. For neutrons of wavelength large compared to nuclear
222 The Neutron [Pt. VII
dimensions, the wave properties are of primary importance in deter
mining the nature of the interaction between neutrons and nuclei.
Since nuclear radii fall in the range 2 X 10~ 13 to 10 12 cm, the wave
properties of the neutron are seen to be important for energies up to
the fast neutron region; for fast neutrons, the wave properties are of
comparatively lesser significance.
The wavelengths of thermal neutrons are of the same order as the
interatomic distances in solid matter. Thus, we would expect neutrons
of these energies to show interference effects in their passage through,
and scattering by, ordered materials. Such interference phenomena were
predicted by Elsasser (E5) and by Wick (W24) in 193637, and soon
indicated by Preiswerk and von Halban (P29) and by Mitchell and
Powers (M33). Indeed, thermal neutrons have been found to behave
very much like xrays under similar circumstances, with the important
difference that the scattering and absorption of neutrons are nuclear
phenomena, whereas the corresponding properties of xrays arise from
their interaction with atomic electrons. Neutron diffraction and inter
ference phenomena will be discussed in detail in Section 5.
It is worth observing that, although wave effects are relatively less
important for fast neutrons, they may possibly become significant again
for ultrafast neutrons. In this energy range the neutron wavelengths
are of the order of the distance between nucleons in the nucleus, and
the scattering from the nucleons within a nucleus can exhibit interfer
ence effects. Indeed, it is possible that, with the development of tech
niques for studying such effects, investigations of the "form factor" in
nuclear scattering may provide important information on nuclear
structure.
7. Classification according to Energy. In the subsequent discussion
we shall find it convenient to refer to neutrons of different kinetic ener
gies according to the following system of classification:
I. Slow neutrons: < E < 1000 ev
II. Intermediate neutrons : 1 kev < E < 500 kev
III. Fast neutrons: 0.5 Mev < E < 10 Mev
IV. Very fast neutrons: 10 Mev < E < 50 Mev
V. Ultrafast neutrons: 50 Mev < E
Although the dividing lines between these categories are quite arbi
trary, this system of classification can be justified on two grounds. In
the first place, the interactions of the different classes of neutrons with
Sec. IB] Properties and Fundamental Interactions 223
nuclei and with matter in bulk involve, in general, different reactions
and types of phenomena. Secondly, the methods of producing and de
tecting the different classes of neutrons are quite different.
(a) Slow Neutrons. The behavior of neutrons in this energy range
has been more extensively investigated than in any of the other ranges.
The interaction of slow neutrons with heavy nuclei is characterized by
sharp absorption resonances and large absorption cross sections for very
lowenergy neutrons. This is also the region in which crystal effects
are important.
Of particular importance to the study of the slow neutron region is
the fact that a number of instruments, known as monochromators, have
been developed whereby neutrons of a given energy may be singled out
and their properties studied. Monochromators in the slow neutron re
gion are characterized by high resolving power, so that the dependence
of a particular effect on the neutron energy can be ascertained to high
accuracy.
The slow neutron range is conveniently subdivided into a number of
subranges, of which the most important are :
(1) Cold Neutrons. These constitute a special category of slow neu
trons of energy less than ^0.002 ev ; they exhibit an anomalously large
penetrability through crystalline or polycrystalline materials.
(2) Thermal Neutrons. In diffusing through materials with relatively
small neutron absorption, slow neutrons tend to assume a velocity dis
tribution of the Maxwellian form
dniv) = Av 2 e M °* /2kT dv (3)
The peak of the Maxwellian distribution is at an energy E — kT, where
T is the absolute temperature of the medium through which the neutrons
are diffusing. Neutrons having such a velocity distribution are referred
to as thermal neutrons.
Frequently the neutrons diffusing through a given medium are not
in thermal equilibrium with the medium. This results in a rather
greater preponderance of highenergy (slow) neutrons than is given by
the Maxwell distribution function. Slow neutrons originating from such
a source, generally of energy above ~0.5 ev, are often referred to as epi
thermal neutrons.
In much of the work on slow neutrons the thermal neutrons are
separated from neutrons of higher energy (say epithermal) by taking
advantage of their strong absorption in relatively thin layers of cad
mium. Those neutrons in a given distribution which are absorbed by
224 The Neutron [Pt. VII
a cadmium layer are sometimes called Cneutrons; the neutrons which
penetrate cadmium (energy > 0.30.5 ev) are, correspondingly, called
epicadmium neutrons.
(3) Resonance Neutrons. This classification refers to slow neutrons
of energy between ~1 and ~100 ev; it is based on the large number of
distinct, sharp, absorption resonances which have been observed in the
interaction of neutrons, in this energy range, with heavy nuclei.
(b) Intermediate Neutrons. In the intermediate range the predomi
nant type of neutron reaction is elastic scattering. Until fairly recently
this region has been the least extensively studied, mainly because of the
lack of suitable neutron sources and detectors. Recently a number of
techniques have been developed for the study of this energy range.
These techniques are now being extensively exploited.
(c) Fast Neutrons. The fastneutron region is characterized by the
appearance of many nuclear reactions which are energetically impossible
at lower neutron energies, of which the most important is inelastic scat
tering. This region has been fairly extensively investigated, although
the available techniques have been rather crude as compared to those
used in the slowneutron range.
(d) Very Fast Neutrons. This energy interval is distinguished from
the preceding by the appearance of nuclear reactions involving the emis
sion of more than one product, such as the (n,2n) reaction. It is a rela
tively unexplored energy range, mainly because of the comparatively
small number of suitable neutron sources.
(e) Ultrafast Neutrons. The development of ultrahigh energy particle
accelerators has resulted in the possibility of producing ultrafast neutrons
and studying their properties. Before the advent of these machines, this
region was accessible only through the utilization of the neutrons in the
cosmic radiation. A distinguishing feature of nuclear reactions in this
energy range comes from the relatively small interactions of neutrons
with nuclei, resulting in a partial transparency of nuclei to ultrafast
neutrons. Also, for these high energies, "spallation reactions" — in
which the bombarded nucleus emits many fragments — are observed.
C. Fundamental Interactions
Of primary importance for the understanding of nuclear forces is the
study of the interaction between neutrons and other nucleons — neutrons
and protons — and between neutrons and the lighter particles (electrons,
mesons) . There have been many experimental and theoretical investiga
tions which shed light on the nature of nuclear forces. Among the most
important are the scattering of neutrons (and protons) by protons, the
Sec. 1C] Properties and Fundamental Interactions 225
capture of neutrons by protons, the static and dynamic properties of
the deuteron, the scattering of neutrons by hydrogen molecules, and
some of the properties of heavy nuclei. Since most of these results have
already been discussed in Part IV of Volume I, they will merely be
presented here in summary form.
However, since a number of aspects of the neutronproton interaction
are of primary importance for the understanding of the interaction of
neutrons with matter, such aspects will be discussed more fully. Thus,
the scattering and capture of neutrons by protons will be covered in
some detail. The discussion of the scattering of neutrons by hydrogen
molecules, involving as it does the wave properties of neutrons, will be
reserved for Section 5.
1. The NeutronProton Interaction, (a) Properties of the Deuteron.
(1) Static Properties. The ground state of the deuteron is essentially a
3 S% configuration. 1 From the known binding energy of the deuteron,
conclusions can be drawn concerning the force (potential) between the
neutron and proton in this state, in which the neutron and proton spins
are aligned in the same direction (total spin = spin of the deuteron
= 1). The static properties of the deuteron give no information con
cerning the neutronproton potential in the singlet state (total spin = 0)
(B22).
(2) Photodisintegration. The deuteron can be disintegrated into a
neutron and a proton by the absorption of an amount of energy greater
than its binding energy, 2.23 Mev. Although this can be accomplished
through the bombardment of deuterons by a variety of particles, the
disintegration of the deuteron was first observed (C8) in bombardment
by gamma radiation. This process, the socalled photodisintegration
of the deuteron, has been most extensively investigated, and has yielded
much useful information concerning the neutronproton interaction
(B22).
Immediately after its discovery, Bethe and Peierls (B20) and Massey
and Mohr (M6) propounded the theory of the disintegration of the
deuteron by a photoelectric effect. Soon afterward, Fermi (F15) sug
gested a second mechanism which contributes to the disintegration: a
photomagnetic process. In the photoelectric disintegration the electric
field of the gammaradiation acts on the instantaneous electric dipole
moment of the deuteron (the average electric dipole moment is, in the
1 The deuteron has a small electric quadrupole moment, which fact implies that
the groundstate wave function contains, in addition to the predominant 3 Si func
tion, a small admixture of 3 Z>i form. The implications of this fact, especially with
regard to the tensor nature of nuclear forces, has been discussed in detail in Part IV
of Volume I.
226 The Neutron [Pt. VII
ground state, zero). After separation of the proton from the neutron,
through the action of the electric field, the spins of the neutron and
proton remain parallel; thus, the photoelectric effect involves only the
properties of the neutronproton interaction in the triplet state. In the
second process the magnetic field of the radiation interacts with the
magnetic dipole moments of the neutron and proton, the effect of the
interaction being to "flip" the spin of one of the particles with respect
to that of the second; thus, the photomagnetic process also involves the
properties of the neutronproton potential in the singlet state.
The contributions of the two processes can be separated experimen
tally, since they result in different angular distributions of the emerging
particles with respect to the direction of the incident gammaray. These
angular distributions can be understood in terms of a rough, semiclassi
cal model of the deuteron as a neutron and proton separated by a fixed
distance and having equal probability for all possible orientations of the
connecting line. Thus, in the case of photoelectric disintegration, the
effect is greatest when the deuteron's dipole moment is in the direction
of the electric field vector of the gammaray beam, which is perpendicu
lar to the direction of motion of the photon. This leads to a distribu
tion of recoil neutrons and protons (in the centerofmass coordinate
system) proportional to the square of the sine of the angle between the
incident photon and the recoils.
In the photomagnetic disintegration, on the other hand, the effect is
produced by the difference between the action of the photon's magnetic
field on the neutron and the proton, which is due to the difference be
tween the magnetic dipole moments of the two particles [—1.91280 and
+2.79255 nuclear magnetons, respectively (Ml)]. This difference is
independent of the orientation of the deuteron, and hence the recoils,
which are due to the photomagnetic disintegration, are spherically sym
metrically distributed in the centerofmass coordinate system. 1
In addition to the different angular distributions, the cross sections
for the photoelectric and photomagnetic disintegrations have different
dependences on the gammaray energy. The cross sections start out at
zero at the photodisintegration threshold, rise to maxima for values of
1 This difference in the angular distributions of the recoil products follows from
simple quantummechanical considerations. Thus, in the photoelectric effect the
final state of the separated neutron and proton must be a P state to satisfy the
selection rule for electric dipole radiation, Ah = ±1; also, since the electric field
does not act, in first order, on the nucleon spins, AS  0. The transitions occur
from the state of the deuteron for which m = ±1 (with respect to the direction of
the incident photon); hence the sin 2 distribution of the recoils. Correspondingly,
the magnetic dipole transition requires AL = 0, AS = ±1. Thus, the final state is
a 1 S a state, with a spherically symmetrical distribution of the recoils.
Sec. 1C] Properties and Fundamental Interactions 227
E y = — 2.2 Mev of approximately twice the binding energies (of the trip
let and singlet states, respectively), and then fall off with increasing
gammaray energy. 1 For photon energies close to the threshold, the
two effects are comparable in importance. For photon energies large
compared to the binding energy of the deuteron, the photomagnetic
effect is negligible compared to the photoelectric effect. Thus, the
angular distribution of the recoils can be described by the expression
a + b sin 2 0, with a and b both energydependent, and a falling off much
more rapidly than b with increasing gammaray energy. For photon
energies considerably above the threshold, the simple considerations
described above are no longer completely applicable; other effects, such
as the tensor nature of the neutronproton interaction, come into
play (R6).
Since the early attempts to observe the cross section and angular
distribution of the recoils in the photodisintegration of the deuteron
(CIO), the techniques of measurement have been considerably improved.
More recent results (B30, G28, W44) confirm the theoretical predictions
and provide important information concerning the nature of the neutron
proton interaction.
(6) The Capture of Neutrons by Protons. Although, for neutrons of
most energies, the most important effect in the neutronproton interac
tion is the scattering process, discussed in (c) below, it was soon observed
(W18, D19, A20) that neutrons are appreciably absorbed in hydrogen
containing substances. This absorption is attributable to a radiative
capture (n,y) process, a reaction to be expected, since the combination
of a neutron and a proton into a deuteron is energetically preferred.
The gammarays accompanying this process were first observed by Lea
(L10). The radiative capture reaction, n + p — > d + 7, is the inverse
of the photodisintegration of the deuteron, and the calculation of its
cross section follows directly from the photodisintegration calculations
(B22).
Such a calculation indicates that capture of a slow neutron by a photo
electric process (emission of an electric dipole gammaray) has a negligi
ble cross section (B20) ; this result is easily understood, since photoelec
1 If (3 = 2.2 Mev and  n  = 0.065 Mev are the absolute values of the binding
energies in the triplet and singlet states (the singlet state is actually unbound) and
E = E y — €3, then the simple theory gives (B22)
&re 2 h 2 t3 A E*
ff mag —
3 he M (E + e 3 ) s
2ir e 2 h 2 Qfr  ;Q 2 («* + 1 q *)»«,*g*
3 he M Mc 2 (E + caKE + ta I)
228 The Neutron [Pt. VII
trie capture requires that the neutronproton system be, initially, in a
P state, which is highly improbable for a slow neutron. Indeed, it was
the observation of an appreciable neutronproton capture cross section
which led Fermi (F15) to postulate a photomagnetic capture process
and its inverse, the photomagnetic disintegration. Since the photo
magnetic capture involves only S states CS —> s Si), the resulting
cross section has al/» dependence on the neutron energy and therefore
becomes quite appreciable for thermal neutrons. Fermi showed that
the photomagnetic capture is indeed strong enough to account for the
relatively rapid absorption of thermal neutrons in hydrogenous materials.
The precise experimental determination of the neutronproton (n,y)
cross section is difficult, since capture represents only a small fraction
of the total neutronproton cross section at even the smallest available
neutron energies and, further, it leads to a nonradioactive end product.
Thus, none of the usual methods of measuring cross sections (Section 3)
is easily applicable. Its value can, however, be inferred from measure
ments involving the diffusion of thermal neutrons in hydrogencontain
ing materials (Section 4), since the average distance traveled, before
absorption, by a thermal neutron depends on the capture as well as on
the scattering cross section (A20). Or, equivalently, the mean life of a
thermal neutron in a hydrogenous material is inversely proportional to
the capture cross section, so that a measurement of the mean life can
yield a value for <r(n,y) (A18, M4).
The most accurate value of the (n,y) cross section has been obtained
from a direct comparison of slow neutronproton absorption with the
absorption of boron, which also has a 1/v energy dependence (R8, W19).
Whitehouse and Graham (W19) obtained
— = 2270 ± 68
Using the value <tb = 710 ± 21 barns for neutrons of velocity v =
2200 m/sec (R20), this yields
<m(n,y) = 0.313 ± 0.013 barns
Since the photomagnetic capture and disintegration processes involve
the properties of the neutronproton interaction in the singlet state, the
magnitudes of the cross sections can be used to infer the strength of this
interaction. In particular, different crosssection values are predicted,
depending on whether the singlet state is bound or unbound (B22). In
the case of the photodisintegration experiments, the crosssection deter
minations are not sufficiently accurate to allow an unambiguous choice
between the two possibilities (G28).
Sec. 1C] Properties and Fundamental Interactions
229
The neutronproton radiative capture cross section is measured with
sufficient accuracy. Rosenfeld (R19) calculates the cross section, for
neutrons of energy 0.026 ev, to be 0.32 or 0.16 barn for the cases, respec
tively, of an unbound or a bound singlet state. The experiments
strongly favor the conclusion that the ^o state of the deuteron is un
bound. "But [chiefly because of the neglect of the exchange effect] the
calculation is not sufficiently accurate to allow us to regard this evidence
as entirely conclusive."
The conclusion is, however, completely borne out by the measure
ments on the scattering cross sections for ortho and parahydrogen (see
Section 5).
(c) The Scattering of Neutrons by Protons. (1) Cross Section. Since
the first investigations of Chadwick (C7), there have been many meas
urements, at various neutron energies, of the cross section for the scat
tering of neutrons by protons. The results are summarized in Fig. 2,
_, 2
M
a
u
t S
b 6
4
H
0.03
23468 23468 28468 23468 2
0.01 0.1 1 10 100 300
E n (Mev)
Fig. 2. Cross section for the scattering of neutrons by free protons vs. kinetic energy
of the neutron, in the range 0.01300 Mev, from the compilation of Adair (A2).
Additional data, in the ultrafast range, are given in (T3) and (D12).
in which the neutronproton scattering cross section is plotted as a func
tion of the neutron energy, between 0.01 and 300 Mev.
The quantummechanical treatment of neutronproton scattering was
first given by Wigner (W27), who showed that the cross section as a
230 The Neutron [Pt. VII
function of neutron kinetic energy E is given (in the limit of neutron
wavelength large compared to the range of the neutronproton force) by
the expression „
4irh 2
er(np) = ■ = : — j (4)
M(^+c)
where e is the binding energy of the neutronproton system (deuteron).
The above expression must be corrected for the finite range of the neu
tronproton force (B22); however, this correction is independent of E
and need not be considered for the purposes of the arguments which
follow. The above expression was found to agree with the observed
cross sections for neutrons of ~1 Mev and greater.
When applied to slow neutrons, the above expression yields a value
of ~3 barns. The early experiments (D19) gave a value, very much
greater, of ~35 barns. (The cross section for the scattering of neutrons
by free protons has, in fact, been determined to be ~20 barns.)
This serious discrepancy was explained by Wigner x under the assump
tion that the force between a neutron and proton depends on the rela
tive orientation of their spins. Since in a fourth of the cases the neutron
proton scattering takes place in the singlet state, the cross section should
be given by the expression
47rh 2 /3 1 1 1 \
c(np) = 1  i r \ — ^ i r ) (5)
M \4^+6 3  4tf+U/
where e 3 and « x refer, respectively, to the binding energies of the triplet
and singlet states of the deuteron. To fit a slowneutron cross section
of 20 barns,  ei  = 0.065 Mev.
From the neutronproton scattering, it is impossible to determine the
sign of €i. Actually, we know from other evidence that the singlet state
of the deuteron is unbound. Thus, although the term "binding energy"
has no meaning for this state, «i represents a certain combination of the
constants which describe the neutronproton interaction in the singlet
state (B22).
Early measurements of the slow neutronproton scattering cross
section (D19, W18, A20) showed a wide variation, outside the experi
mental uncertainties. The reason for these variations was given by
Fermi (F17), who ascribed them to the effect of the binding of the pro
1 Although it is universally acknowledged throughout the literature (F5, B22) that
Wigner originated the idea of the spin dependence of the neutronproton force to
explain the large neutronproton scattering cross section for slow neutrons, he has
not, to our knowledge, published these considerations.
Sec. 1C] Properties and Fundamental Interactions 231
tons in the material — molecule, liquid, or solid — used as a neutron scat
tered Such effects are, of course, negligible for neutron energies con
siderably greater than the binding energy of the proton in the system
under consideration. They are most important for neutron energies
less than the lowest excitation energy of the system in which the proton is
bound. The proton behaves like a particle of infinite mass when bound
in solids or liquids, or like a particle of the total mass of the molecule
involved in the case of neutron scattering by gaseous materials. It can
be shown that the scattering cross section of slow neutrons by bound
protons varies directly as the square of the reduced mass of the neutron,
the scattering being spherically symmetrical in the centerofmass co
ordinate system. Thus, the scattering cross section of infinitely slow
(zero energy) neutrons by protons in solids or liquids should approach
four times the free proton cross section, or about 80 barns.
The variation of the neutronproton cross section in the energy range
in which the proton is neither free nor completely bound (energy greater
than the lowest excitation energy but less than the binding energy of the
protons) depends on the specific properties of the protoncontaining
material under consideration and is rather difficult to compute. Dis
regarding interference effects due to the crystalline properties of the
solid under consideration or to the order introduced by the molecular
structure (see Section 5), the cross section can be shown to decrease
monotonically from the bound proton to the free proton cross section
as the neutron energy increases from to several electron volts (A31,
M27).
Thus, the free proton scattering cross section for slow neutrons can
be obtained from measurements on neutrons in the resonance energy
range. As a result of such measurements (C18), the slow neutronfree
proton scattering cross section was determined to be slightly less than
21 barns. However, binding effects still have a small influence for neu
trons with energy of a few electron volts. In the most recent experi
ments (M24), the neutronproton cross section has been measured as a
function of the neutron energy, and theoretical considerations (P19)
were used to extrapolate to a free proton neutronproton scattering
cross section of 20.36 ± 0.10 barns for neutrons of zero kinetic energy.
Effects of nuclear binding are, of course, present in the scattering of
neutrons by heavier nuclei, although their importance decreases rapidly
with increasing mass number A of the scattering nucleus. In general,
for nuclei bound in solids or liquids,
(A + 1\ 2
Cbound — I I ""free (6)
232
The Neutron
[Pt. VII
Thus, for deuterons bound in solid material, <rb OU nd = 2.25<rf re e, while,
for the scattering of slow neutrons by carbon, abound = 1.17of ree .
(2) Angular Distribution. It was first pointed out by Wigner (W27)
and by Wick (W22) that the angular distribution, in the scattering of
20 
15 
slO
1 1 1 1
1
1 ' ' 1
o
"""»>.
„ *■"'
V N
°s'
v s 40 Mev
s
s
\
V
\
/
\
/
\
/
\
fa °
\
' T
\
•
v
/
— s
s
° o
o"
"'o'
s'o °
=~^ n 90 Mev
s
A '
N T
 — tk~
\ / \
.7 V
y
j.
1 1 1
\260 *Xi
\Mev £4rj T
 \ S^.x
h ■■
1 1 1 1
5
.y
i
20
40
60 80 100 120 140 160 180
Scattering angle
Fig. 3. Angular distributions in the scattering of ultrafast neutrons by protons.
The figure is from a paper by Kelly, Leith, Segre, and Wiegand (K9).
fast neutrons by protons, should provide information concerning the
neutronproton interaction in states of higher angular momentum (P, D,
etc). The scattering of slow neutrons by protons involves only the
state of zero angular momentum (S state) and is therefore spherically
symmetrical in the centerofmass coordinate system. The energies at
which higher angular momentum scatterings set in measure the strengths
Sec. 1C] Properties and Fundamental Interactions 233
of the interaction in these states, while the form of the angular distribu
tion leads to inferences concerning the nature of the potential.
A number of experiments using d + d neutrons (2.4 to 2.7 Mev) show
that, at these energies, the neutronproton scattering is spherically sym
metrical in the centerofmass system (R19). While the scattering is
still essentially spherically symmetrical at 14 Mev (BIO), some of the
experiments in the 914Mev neutron energy range (L7, P24) show indi
cations of a slight asymmetry, corresponding to a preferential scattering
of protons in the forward direstion. This asymmetry is quite pro
nounced for neutrons of energy 27 Mev (B71). Such an effect is to be
expected on the basis of a theory of a neutronproton interaction of the
exchange type; exchange forces are required to explain the "saturation"
of nuclear forces, since they lead to a repulsive force at short distances
and thus tend to prevent the collapse of heavy nuclei and to maintain
a constant density of nuclear matter.
Neutronproton scattering experiments with 40, 90, and 260Mev
neutrons show large deviations from spherical symmetry; their results
are plotted in Fig. 3. The most interesting characteristic of these re
sults is the rough symmetry about 90° (in the cm. system) of the dif
ferential neutronproton scattering cross section. Serber has pointed
out that these results indicate a neutronproton interaction involving
approximately equal proportions of forces of the ordinary and exchange
type.
Christian and Hart (CI 5) have discussed the type of neutronproton
interaction which is required to fit both the lowenergy and the high
energy data. They use an interaction potential of the Serber type,
choosing the radial dependence to obtain the best possible fit, and also
taking into account the tensor form of the interaction, required by the
electric quadrupole moment of the deuteron.
The Serber interaction, a mixture of equal parts ordinary and exchange
forces, has the property that all terms which correspond to odd orbital
angular momentum quantum numbers vanish; the observed symmetry
of the highenergy neutronproton scattering about 90° is, of course,
the decisive evidence in favor of the Serber interaction, since any appre
ciable interaction in a state of odd orbital angular momentum would
destroy this symmetry. However, the terms of odd angular momentum
are precisely those which lead to a strong repulsion between the neutron
and proton for small distance of separation and, therefore, are required
for the "saturation" of nuclear forces. Indeed, it appears doubtful
whether a nucleonnucleon interaction of the Serber type is consistent
with the saturation of nuclear forces in heavy nuclei.
234 The Neutron [Pt. VII
2. The NeutronNeutron Interaction, (a) Experimental Evidence.
(1) Low Energies. Since the highest available neutron densities are
still far from appreciable (below 10 10 neutrons/cm 3 ), it is manifestly im
possible to obtain information about the neutronneutron interaction
through observation of the scattering of neutrons by free neutrons; our
knowledge of the force between two neutrons must, perforce, be inferred
from other information. In considering the available information, it is
useful to bear in mind the following general considerations: (1) Experi
ments involving neutrons of kinetic energy below ~20 Mev yield infor
mation only about the neutronneutron forces in the singlet S state since,
at these energies, states of higher angular momentum are not appreciably
excited, and the exclusion principle prevents the neutronneutron system
from existing in a triplet S state. Experiments with ultrafast neutrons
can, on the other hand, yield information about the neutronneutron
interaction in both the singlet and triplet states. The same arguments
hold, of course, for the protonproton interaction. (2) Since there is
some reason to expect that the purely nuclear interaction between two
nucleons should be independent of whether they are neutrons or pro
tons (see further on), the evidence on the protonproton interaction can
be considered to have some bearing on the problem of the neutron
neutron interaction. In any event, we shall herein review, briefly, the
available information on the interaction between two nucleons, irrespec
tive of their charge. 1
The strongest evidence concerning the charge symmetry of nuclear
forces (equality of the nuclear neutronneutron and protonproton
forces) is derived from the observed energy differences between mirror
image nuclei (nuclei which can be obtained from one another by inter
changing the neutrons and protons). The energy differences between
such nuclei, as measured by the maximum energy of the betarays
emitted in the decay of one of the pair into the other, can be completely
accounted for by the neutronproton mass difference and the difference
in the electrostatic energies of the two nuclei. In addition, mirror nuclei
are similar with respect to their observed level structure. From this
evidence, we conclude that the purely nuclear neutronneutron and
protonproton forces are essentially equal. Actually the evidence on
nuclear level structure lends strong support to the stronger hypothesis
of charge independence (equality of the neutronneutron, neutron
proton, and protonproton forces in the same states), since nuclear iso
bars are observed to exhibit markedly similar level patterns.
1 A much more complete discussion of these points is contained in Part IV of Vol
ume I,
Sec. 1C] Properties and Fundamental Interactions 235
The purely practical restrictions, which prevent the performance of
neutronneutron scattering experiments, do not obtain in the case of
protonproton scattering. Many such experiments have been performed
for proton energies up to <~15 Mev. From these it can be concluded
that within the experimental uncertainty (and taking into account the
effect of the Coulomb interaction) the nuclear neutronproton and pro
tonproton forces in the 1 S state appear to be equal.
Thus, the available evidence supports the idea of the charge inde
pendence of the nuclear forces between two nucleons.
(2) On the Possible Existence of a Stable DiNeutron. The strength
of the neutronneutron interaction in the '(So state determines whether
or not there can be a stable system consisting of two neutrons. (There
is no possibility for a stable twonucleon state of higher orbital angular
momentum.) The existence of such a stable "dineutron" is, until now,
experimentally neither proved nor disproved. If the nucleonnucleon
interaction is completely charge independent, the dineutron will be
unstable, since the '(So neutronproton system is unstable by about
65 kev; however, this is so close to being stable that it would require
only a rather small increase of the neutronneutron interaction over the
corresponding neutronproton interaction to lead to a stable dineutron.
A stronger argument against the existence of a stable dineutron can
be derived from the equality of the lowenergy neutronneutron and
protonproton forces, as evidenced by the properties of the mirrorimage
nuclei. Analysis of lowenergy protonproton scattering (Jl) proves,
conclusively, that the '£ protonproton interaction is not strong enough
to lead to a bound state (even in the absence of the Coulomb repulsion).
Although the weight of available evidence does not favor the existence
of a stable dineutron, it is, nevertheless, of interest to consider some of
the possible consequences of its existence. Feather (F4) has pointed
out that it is possible to obtain an upper limit to the binding energy of
the dineutron from the observation that the nucleus He 6 does not decay
according to the reaction He 6 »• He 4 + n 2 . From the masses of the
nuclei involved, he concludes that the binding energy of the dineutron
is less than 0.7 ± 0.2 Mev. He also points out that the dineutron
would be betaunstable, n 2 » H 2 + P~ + v, with a mean life (assum
ing an allowed transition) of 1 < t < 5 sec.
If the dineutron should exist as a stable configuration, and if it should
be possible to obtain it in sufficient numbers, it could easily be detected
in experiments involving scattering by nuclei. For the two neutrons
would be scattered coherently, and the interference effects would per
sist to considerably higher energies than do the normal coherence effects
in slowneutron scattering, since the dineutron would be a compact sys
236 The Neutron [Pt. VII
tem, of dimensions ~10 12 cm. Thus, the scattering of dineutrons by
hydrogen atoms would be the same as the scattering of cold neutrons by
parahydrogen molecules (see Section 5), for which the cross section is
~4 barns, as compared to ~20 barns for the free neutronproton scat
tering cross section.
Furthermore, as pointed out by Feather, capture of a dineutron
would, as compared to ordinary neutron capture, result in a different
compound nucleus and in characteristic radioactivities.
The difficulty in observing such effects would, if the dineutron were
stable, arise from the difficulty of obtaining an appreciable source of
dineutrons. Such a source would have to be obtained from a suitable
«.uclear reaction, in which dineutrons are emitted. It could not be ob
tained by neutronneutron collisions in a region of high slowneutron
density (even if neutron densities of sufficient magnitude were avail
able) because the required reaction, n + n — > n 2 + 7, would be highly
forbidden, since it would involve the transition 1 S — * 1 £>o Although
a dineutron could, if stable, be produced in a collision involving three
neutrons, such collisions would be exceedingly improbable.
Fenning and Holt (Fll) have attempted to detect the presence of
dineutrons in the Harwell pile, by looking for alphaparticles from the
Bi 209 + n 2 > AcC 211 — ^ AcC" ?> Pb 207
2.16 min 4.8 min
(They were investigating the possibility that dineutrons might be emit
ted in slowneutron fission.) They exposed bismuth to a flux of ~10 12
neutrons cm 2 sec 1 , but could detect no activity attributable to
AcC, from which they established an upper limit of 1.5 X 10 21 sec 1
for the product of the dineutron flux and the cross section for its absorp
tion in bismuth.
In considering the possible effects of the dineutron on nuclear reac
tions, it should be noted that such effects might exist even if the di
neutron is not stable, for the dynamics of a reaction involving two neu
trons would be quite different if the two neutrons were absorbed or
emitted as a single unit than if they behaved independently (C19).
Thus, Kundu and Pool (K25) consider the characteristics of the excita
tion of (H 3 ,p) reactions on rhodium and cobalt as "probable evidence of
the dineutron." Another, and more favorable, reaction in which effects
of a dineutron might be observed involves the two possibilities
H 3 + H 3 » He 4 + 2n + Q t
> He 4 + n 2 + Q 2
If the two neutrons are emitted independently, their energies, and that
Sec. 1C] Properties and Fundamental Interactions 237
of the recoil alphaparticle, will vary over a rather wide range (for a
given energy of the bombarding triton), as is to be expected in a three
particle reaction. If, on the other hand, the two neutrons are emitted
as a single particle, the energies of the dineutron and of the alpha
particle are uniquely determined, for a given angle of emission, by the
laws of conservation of energy and momentum. The weight of available
evidence favors the first of the two abovementioned reactions (L22,
A9, L14).
Perhaps the most striking evidence concerning the dineutron is
derived as a byproduct of the experiments of Panofsky, Aamodt, and
Hadley (P3) on the absorption of negative pimesons by deuterium in
the reaction ir~ + d — » In + y. In interpreting the observed gamma
ray energy distribution, Watson and Stuart (W6) point out that the
shape is strongly dependent on the degree of correlation in the direc
tions of emission of the two neutrons; indeed, if the neutrons were always
emitted as a dineutron, the gammaray spectrum would be monochro
matic. The observations are sufficiently accurate to show that there is
a strong interaction (attraction) between the two neutrons; they are not
sufficiently precise, as yet, to allow more than an upper limit of <~200
kev to be placed on the binding energy of the dineutron. The proper
ties of the mesonproducing reaction, 7 + d — » 2n + x + , might also
throw some light on the dineutron.
(3) High Energies. For neutrons with kinetic energies in the ultra
fast range, it becomes possible to observe the neutronneutron interac
tion somewhat more directly in the scattering of neutrons by deuterium
nuclei. If the wavelength of the neutron is much less than the average
separation between the neutron and proton in the deuteron, the two
nucleons should scatter neutrons independently; the cross section for
the scattering of neutrons by neutrons, a(nn), should then be given by
the difference between the scattering cross section of deuterium and that
of hydrogen, ,_
e{nn) = <j(nd) — <r(np) V)
Unfortunately, there are still appreciable effects of the structure of
the deuteron in the scattering of neutrons of a few hundred Mev, since
the neutron wavelength is not negligible compared to the internucleon
distance. Another way of expressing this difficulty is to note that the
nucleons in the deuteron are moving with kinetic energies of internal
motion as high as <~25 Mev; the velocity, corresponding to this kinetic
energy, is only ~i of the velocity of a 100Mev neutron.
A possible way out of this difficulty has been suggested by Segre. 1
Since it is feasible to measure, directly, both a(np) and <r(pp), a meas
1 Private communication.
238 The Neutron [Pt. VII
urement of o(prf) should determine the magnitude of the effect, on the
cross section, of the structure of the deuteron. This correction could
then be applied to the measurement of o(nd) in order to extract the
neutronneutron cross section.
The interpretation of neutrondeuteron scattering for ultrafast neu
trons has been considered by a number of authors, e.g., De Hoffman
(D8), Gluckstern and Bethe (G9), and Chew (C14). It turns out that,
in addition to the expected contributions from neutronproton and
neutronneutron scattering (both averaged over the momentum dis
tributions of the nucleons in the deuteron), there is a cross term due to
interference between the two scattering centers and depending in de
tail on the nature of the interactions. As a result, the simple additivity
relationship of Eq. (7) cannot even be applied at energies of a few hun
dred Mev.
The most extensive attempt at interpretation of neutrondeuteron and
protondeuteron scattering experiments in the ultrafast energy range
has been carried out by Chew (C14). He has treated the problem by
the "impulse approximation," in which the scattering nucleon is as
sumed to behave in a relatively independent fashion during the time of
the impact. This approximation appears to be appropriate for the
treatment of some aspects of the neutrondeuteron scattering problem,
e.g., the dissociation, by neutron impact, of the deuteron; however,
other aspects, such as elastic neutron scattering or proton "pickup,"
cannot be treated in this fashion. From the available data, Chew con
cludes that there is "no evidence for a difference between neutron
neutron and protonproton interactions."
The results of measurements, on the cross sections for scattering of
ultrafast neutrons by protons and deuterons, are summarized in Table 2. 1
TABLE 2
Total Cross Sections of Protons and Deuterons for Ultrafast Neutrons
cr(nd) <rinp) Difference
E n (Mev) (barns) (barns) (barns) Reference
42 0.289 ±0.013 0.203 ± 0.007 0.086 ±0.015 H61
85 0.117 ±0.005 0.083 ±0.004 0.034 ±0.006 C25
95 0.104 ±0.004 0.073 ±0.002 0.031 ± 0.004 D9
270 0.057 ±0.003 0.038 ± 0.002 0.019 ±0.003 D10
280 0.049 ±0.005 0.033 ±0.003 0.016 ± 0.006 F41
(b) Some Comments Pertaining to the Meson Theory of Nuclear Forces.
On the basis of the observed properties of the forces between two nu
cleons (in particular, the short range), Yukawa predicted the existence
1 For results of pp scattering measurements see Volume I, Part IV, Section 2A.
Sec. 1C] Properties and Fundamental Interactions 239
of a particle of mass intermediate between that of the electron and the
proton. According to his idea the force between two nucleons is regarded
as resulting from the interchange of such particles, called mesons, be
tween nucleons, in much the same way that purely electrical forces may
be regarded as resulting from the interchange of photons between
charged particles. There have been developed, since Yukawa's sugges
tion, a number of different types of meson theories of nuclear forces;
these differ in the properties of the assumed mesons and in the type and
strength of the assumed coupling between the meson and the nucleon
fields (P7).
Yukawa's prediction was soon followed by the discovery of such a
particle in the cosmic radiation. This particle, the mumeson or muon,
has a mass of ~210 electron masses, may be either positive or negative,
and is unstable, decaying into an electron and two neutral particles of
negligible rest mass (presumably neutrinos) with a mean life of 2.15
X 10 — 6 sec. However, subsequent investigation has shown that the
muon has a very weak interaction with nucleons, a fact which eliminates
it from the role of nuclear binding material.
There is, however, another type of meson, the pimeson or pion, which
appears to be more directly associated with the nuclear forces. It has
a mass of ~275 electron masses, and decays into a muon and (presum
ably) a neutrino, with a mean life of 2.6 X 10~ 8 sec. Pions have been
produced in the interaction between two nucleons, or between a nucleon
and a gammaray, when there is enough energy available to supply the
rest mass. In addition to charged pions of both sign, there have also
been observed neutral pions (whose mass appears to be slightly less
than the mass of the charged pion) which decay into two gammarays,
with a very short mean life (~10 — 15 sec). Finally, there is evidence
for the existence of one or more types of still heavier meson. Very little
is known concerning their interactions with nuclei.
Of the various types of mesons which have been observed, it is the
pion which most probably plays the role of nuclear binding agent. It is
produced directly in the interaction between two nucleons, and between
nucleons and gammarays; the cross section for its production is of an
order of magnitude which indicates that it interacts strongly with nu
cleons — strongly enough to account for the strength of nuclear forces.
Indeed, recent work on the production of pions, through the use of
ultrahighenergy machines, and on the interaction of pions with nucleons
and nuclei provides the strongest arguments for the fundamental valid
ity of the meson theory of nuclear forces.
Although meson theories have, so far, been unable to account in detail
for the nuclear forces, they do provide a qualitative basis for under
240 The Neutron [Pt. VII
standing the properties of nucleons (like, say, the anomalous magnetic
moments of the neutron and proton) and of nuclear forces. The evi
dence on meson production at high energies together with the evidence
on the charge independence of the nucleonnucleon forces favor a
symmetrical meson theory, in which charged mesons of both sign, as
well as neutral mesons, play a comparable role. The exchange of charged
mesons leads to the neutronproton force, while neutral mesons are ex
changed in the neutronneutron and protonproton interactions.
3. The NeutronElectron Interaction. From the observation that the
neutron has an intrinsic magnetic moment, it may be inferred that the
neutron is a complex particle whose structure contains, at least for part
of its existence, some moving charges. Indeed, according to current
meson theories, the neutron can dissociate, spontaneously, into a
tightly bound system of proton and negative pion, according to the
reaction
n +±p + 7T —
During the fraction of its existence as a neutron proper, it can be re
garded as truly neutral; in the protonpion state, however, the neutron
has a charge structure corresponding to a relatively concentrated posi
tive charge surrounded, at a somewhat greater distance (of the order of
a few times 10 — 13 cm), by a cloud of equal negative charge. The frac
tion of the time during which the neutron is in this dissociated state is
variously estimated in the different meson theories to lie somewhere in
the range ~0.1 to 0.5.
As a consequence of this structure, there is an electromagnetic inter
action between neutrons and electrons, the predominant aspects of
which can be described in terms of the interaction between two magnetic
dipoles (that of the electron and that of the neutron). This magnetic
dipoledipole interaction is, of course, strongly dependent on the rela
tive orientations of the spins of the particles. It has been investigated
extensively in studies of the scattering of slow neutrons by paramagnetic
and ferromagnetic materials; these phenomena will be discussed in de
tail in Section 5.
A magnetic interaction of the dipoledipole type would, because of its
spindependent character, vanish for the case of a neutron interacting
simultaneously with an even number of electrons which are in a 1 »So
state. Nevertheless, it has been observed that there is a small, spin
independent neutronelectron interaction. Such an interaction follows
from mesontheoretic descriptions of the structure of the neutron, and
its observed magnitude is in rough agreement with theoretical expecta
tions (to be discussed later in this section). The strength of the inter
Sec. 1C] Properties and Fundamental Interactions 241
action is such that it leads to a (spinindependent) cross section for the
scattering of neutrons by a single bound electron of ir,~5X 10~ 7
barns.
Although the value of <r e is exceedingly small as compared to the cross
sections for scattering of neutrons by nuclei (~ a few barns), effects of
this scattering are, nevertheless, detectable in the scattering of slow neu
trons by some heavy atoms. The possibility of observing such effects
was first suggested and discussed by Condon (C24). This possibility
results from the interference between the elastic scattering of sufficiently
slow neutrons by the nucleus and by all the electrons in the atom.
Let us consider the scattering of a beam of slow neutrons, wavelength
of the same order as the atomic dimensions, by a single atom containing
Z electrons. The differential scattering cross section is
£ =  a + Za e F(6) \ 2 (8)
where «o an d a e are, respectively, the amplitudes for scattering of a neu
tron by the nucleus and by a single bound electron, 1 i.e., <r e = 4ira e 2 .
F(d) is the atomic electron form factor, which can be computed from the
electron distribution or, alternatively, obtained from data on the scat
tering of xrays (C22). Since c — 4ira 2 5s> <r e , we may write
For neutrons of wavelength large compared to the atomic dimensions,
F(0) — > 1, while, as the neutron wavelength becomes small compared
to the distance between the electrons, F(6) — > 0. To observe a possible
neutronelectron interaction of the magnitude expected, it is absolutely
necessary to take advantage of the interference between the neutron
scattering by the nucleus and that by the atomic electrons.
For an estimate of the magnitude of possible effects, let us assume a
set of arbitrary, but reasonable, values, a ~ 5 barns, cr e ~ 5 X 10~ 7
barns, ZF « 50. Then
— = 2ZF ( — ) ~ 3 percent
00 wo/
1 Strictly speaking, ao refers only to the coherent portion of the nuclear scattering
amplitude. When there is, in addition, an incoherent contribution to the nuclear
scattering, the above expression for the differential atomic scattering cross section
requires the addition of an incoherent term, ai 2 , where the total nuclear scattering
cross section is <r s = 4ir(ao 2 + ai 2 ) Although, for the discussion of this section, we
shall assume that oi = 0, all the expressions to be derived can easily be modified
if the nuclei exhibit any incoherent scattering.
242 The Neutron [Pt. VII
Although it should be possible to measure an effect of this magnitude,
it is necessary to devise some means of distinguishing it from the purely
nuclear scattering, whose cross section is seldom known to an accuracy
of 3 percent. One possibility is the method devised by Fermi and Mar
shall (F23). They compared the scattering of thermal neutrons by
gaseous xenon atoms, into a given solid angle, at angles of 45° and 135°.
Such a difference would arise from two sources: (1) the difference in
the electron form factor which was, for the neutrons employed, F(45°)
— F(135°) = 0.261; (2) an asymmetry in the scattering due to the
centerofmass motion of the scattering system, since the angles were
fixed in the laboratory system. In order to compute the magnitude of
this effect, it is necessary to take into account the thermal motion of
the xenon atoms (Doppler effect), as well as the energy distribution in
the slowneutron beam and the variation of the sensitivity of the de
tector with neutron energy.
Fermi and Marshall observed a definite asymmetry in the scattering
of ~2 percent which, however, became much less than the experimental
uncertainty of 0.85 percent after they applied the Doppler effect cor
rection. Thus, although the accuracy of their measurement is insuffi
cient to confirm the existence of the neutronelectron scattering, their
results set an upper limit to the neutronelectron interaction. If the
interaction is described in terms of a fictitious square well potential of
depth V and range equal to the classical electron radius, r = e 2 /mc 2
= 2.8 X 10~ 13 cm, the experiment of Fermi and Marshall gives  V Q \
<, 5000 ev. (Compare with V <~ 25 X 10 6 ev for the neutronproton
potential.) l
The method of Fermi and Marshall has been reapplied by Hamermesh,
Ringo, and Wattenberg 2 (H28) with, however, significant improve
1 The spinindependent neutronelectron interaction is, if it is of mesonic origin, a
shortrange interaction. Consequently, the value of <r e for slow neutrons depends
only on the volume integral of the interaction potential, rather than on any details
of its shape. The Born approximation yields
AW
flfr
where M is the neutron mass and
f4w
6 = 4x Cv(r)r 2 dr =  (~\
TW
for the fictitious square well potential of range ro. A value of Vo = 5000 ev corre
sponds to a e = 4.3 X 10" 7 barns. It should be noted that a negative b (attractive
potential) corresponds to a negative a,., while the nuclear scattering amplitude, Oo,
is usually positive.
2 I am indebted to Dr. A. Wattenberg for advance communication and discussion
of these results.
Sec. 1C] Properties and Fundamental Interactions 243
ments in the geometry and a large increase in the number of neutrons
counted. They have obtained the value
V = 4100 ± 1000 ev
This value is based on measurements of the scattering of krypton as
well as of xenon; measurements were also performed on argon to check
the correction for the centerofgravity motion. Of the uncertainty of
±1000 ev, quoted by these investigators, half is statistical and half is
their estimate of possible systematic errors. Their result is in good
agreement with the previous measurement of Havens, Rabi, and Rain
water (H46, H47).
The method of observing the neutronelectron interaction, devised
by Havens, Rabi, and Rainwater, is based on observation of the total
cross section
= fda (6) = <r ± 2ZF(cr cr e y A
(9)
They distinguish the effect of the coherent electron scattering by ob
serving the variation of a with neutron energy for wavelengths of the
order of the atomic dimensions. For a neutron source of variable
energy, they employed the Columbia velocity selector; as scattering
materials, they used liquid lead and bismuth. Their method suffers
from the difficulty that it is necessary to apply rather large corrections
to the observed cross section vs. energy curve before the effect of the
neutronelectron interaction can be ascertained. These corrections
arise from a number of causes: (1) In addition to the nuclear scattering
there is, for lead and bismuth, a small nuclear absorption whose cross
section varies with the neutron velocity as 1/v. This correction can be
applied with good accuracy. (2) Since they were dealing with atoms
bound in a liquid, rather than with free atoms, it is necessary to take
into account the effects of the binding (see page 231) on the variation of
the cross section with energy. Although such effects (which include, of
course, a Doppler effect for the motion of the atoms in the liquid) de
pend, to a certain extent, on the details of the binding mechanism, it has
been shown by Placzek (P19) that they can be evaluated with the requi
site accuracy. (3) There are, in the scattering of neutrons by liquids,
effects due to interference between the scattering by different atoms.
Although such effects are relatively small, since the experiments involve
neutrons with wavelengths of the order of atomic dimensions and hence
somewhat smaller than the interatomic distances, they cannot be neg
lected in comparison with the effects of the neutronelectron scattering.
The liquid coherence effects can be shown to vary as X 2 , in the energy
244 The Neutron [Pt. VII
region of interest, and the coefficient of the X 2 term can be evaluated by
a general method for systems of high density and small compressibility,
as shown by the calculations of Placzek, Nijboer, and van Hove (P21).
(4) Although the three effects discussed above, all of which arise from
nuclear causes, can be taken into account in a satisfactory fashion, there
is a possible electronic effect which must be taken into consideration.
Even though the ground states of atomic lead and bismith are 1 S , and
Fermi and Marshall (F23) have proved that scattering due to the neu
tron magnetic moment is negligible for such atoms, small magnetic
scattering effects might possibly be present when these atoms are in the
liquid state.
The results upon which Havens, Rabi, and Rainwater base their
value of the neutronelectron interaction were obtained by careful
analysis of data on the scattering of monoenergetic slow neutrons by
liquid bismuth; they lead to
V = 5300 ± 1000 ev
Of the quoted uncertainty, 650 ev is statistical in origin, the rest being
an estimate of possible systematic errors. 1
Although a detailed mesontheoretic discussion of the spinindepend
ent neutronelectron interaction is outside the province of this review
(even assuming that an adequate theory were available), the phenomena
which can give rise to such an interaction are, however, susceptible of
relatively simple physical interpretation. 2 Actually, there are two dis
tinct types of interaction which, together, can probably account for the
observed effect. The first may be regarded as an electrostatic interac
1 D. J. Hughes has reported (at the February 1952 meeting of the American Phys
ical Society) a measurement of the neutronelectron interaction by observation of
the critical angle for total reflection of cold neutrons from a liquid oxygenbismuth
interface (see Section 5). His preliminary results are in excellent argeement with the
measurements of Havens, Rabi, and Rainwater and of Hamermesh, Ringo, and
Wattenberg. However, this method appears to be capable of yielding the most
accurate value of the strength of the neutronelectron interaction. Using this
method, Harvey, Hughes, and Goldberg have obtained Vo = 4250 ± 400 ev (pri
vate communication, October 1952).
2 Purcell and Ramsey (P32), in their discussion of the possible existence of a neu
tron electric dipole moment, point out that the observed neutronelectron interaction
could, alternatively, be accounted for by the assumption of a small electric dipole
moment of the neutron. The magnitude required to account for the observed inter
action is that of two opposite electron charges separated by a distance of 3 X 10 18
cm. Since the experiments of Purcell, Ramsey, and Smith (S40) have yielded an
upper limit of two electron charges separated by 5 X 10 21 cm for the electric dipole
moment of the neutron, the explanation of the observed neutronelectron interaction
lies, in all likelihood, along the mesontheoretic lines indicated below.
Sec. 1C] Properties and Fundamental Interactions 245
tion which arises from the fact that the neutron is not a strictly neutral
particle but behaves, at least during an appreciable fraction of its
existence, as a positive charge surrounded by a negative meson cloud.
Thus, when the neutron and electron are separated by a distance less
than the extent of the meson distribution (which is of the order of the
classical electron radius r ) they will interact electrostatically. The
predicted magnitude of this effect depends on the form of the meson
theory used in its calculation and, in any event, has only been calcu
lated approximately (i.e., by perturbation techniques, to first order in
the coupling constant). Such calculations have been carried out by
Slotnick and Heitler (S39), by Case (C2), by Dancoff and Drell (D2),
and by Borowitz and Kohn (B48). These calculations predict effects of
the order of, but somewhat smaller than, the observed interaction.
Thus, for instance, Case (C2) obtained an attractive interaction which,
when expressed as a square well potential of range r , yields Vo =
300(/ 2 /2t) ev; f 2 /2w, for this case (the coupling constant for a pseudo
scalar meson with pseudoscalar coupling), is ~5, giving V «* 1500 ev.
The second effect arises, as was pointed out by Foldy (F37), as a
direct consequence of the anomalous magnetic moment of the neu
tron. It can, in fact, be computed, without specification of the form
of the meson theory, 1 by assuming that the neutron obeys the Dirac
equation with the additional "Pauli terms" (P7) in the Hamiltonian,
— n n (eh/2Mc)(JS(r H — iPaE). (n n ^ —1.91 is the neutron moment in
nuclear magnetons.) The <rH term leads to the spindependent, mag
netic dipoledipole interaction between the neutron and the electron. The
second term leads to an interaction of the form — yii„(eh 2 /4M 2 c 2 )8 div E
which corresponds, for the field of a point electron, to an attractive
interaction with
7o= (T)(7) 2 (5) 2 " c2 = 4100ev (10)
Thus, the Foldy term alone appears almost sufficient to account for the
observed interaction. A more accurate measurement of the strength
of the neutronelectron interaction should, indeed, eventually lead to
an evaluation of the magnitude of the electrostatic interaction and
permit a check on the applicability of specific meson theories.
1 Since the fact that the neutron has an anomalous magnetic moment is, itself, a
consequence of meson theories, it could not be said that this effect is of nonmesonic
origin. Indeed, some of the mesontheoretic calculations (S39, D2) include the effect.
However, since no existing theory is capable of yielding the observed value of the
neutron magnetic moment, it appears more reasonable to follow the method of
Foldy, which assumes the observed neutron moment, and to compute this effect by
a phenomenological treatment.
246 The Neutron [Pt. VII
The two effects described above are susceptible of relatively simple,
orderofmagnitude computations. The strength of the electrostatic
neutronelectron interaction has been estimated by Fermi and Marshall
(F23), who considered a model of a point proton (charge g 2 e) surrounded
by a negative meson cloud of charge density
' w (^Mir) <"'
(13)
This expression is suggested by Yukawa's original (scalar) theory of the
meson field, according to which the extent of the meson wave function
is essentially determined from the uncertainty principle,
a ^ h/KC (12)
where k is the pion mass («275 electron masses), g 2 ~ 0.1 to 0.5 is the
is the fraction of time during which the neutron exists in the proton
meson state.
The potential energy U(r) of interaction of the above charge distribu
tion with a point electron can be obtained by solution of the Poisson
equation, and the strength of the neutronelectron interaction is given by
b = 4x f U(r)r 2 dr =  () g 2 e 2 a 2
Assuming a potential well of radius r and depth Vo, we obtain
36 /g 2 \ /m\ 2 /hc\ 2 ,
Considering that the computation is classical and, furthermore, modeled
on a meson theory which is known to be inadequate, the agreement with
the mesontheoretic computations, quoted in the preceding, is quite
satisfactory.
It is somewhat more difficult to give a simple physical interpretation
of the interaction which arises from the neutron's magnetic moment.
However, Weisskopf l has suggested a rather ingenious derivation of
the Foldy term: The assumption that the neutron obeys the Dirac equa
tion leads to a zitterbewegung in the motion of the neutron, such that its
path is a spiral whose radius is of the order of the neutron's Compton
wavelength, R ~ h/Mc. The neutron travels with the velocity of
light, c, in this spiral path whose pitch is such as to give a transport
velocity equal to the velocity v of the neutron. (The intrinsic spin
angular momentum of the neutron can be interpreted as arising from
this spiral motion.) Accordingly, when the neutron is within a distance
1 Private communication.
Sec. 2A1 Interaction with Nuclei 247
R of the electron, which, we assume, behaves like a point charge, there
is a magnetic spin orbit interaction between the electron current and
the neutron's intrinsic magnetic moment whose energy is, neglecting
numerical factors,
'"(s)^^ (15)
Since this interaction has a range of ~R, we obtain
ER 3 /hc\ 2 /m\ 2 ,
T z~*k)Kz) m * (10s>
which is, aside from the factor f, the result of the calculation of Foldy.
Since the interaction depends only on the electron's charge, it is easily
seen to be spinindependent.
SECTION 2. INTERACTION WITH NUCLEI
A. Introduction
The experiments through which the existence of the neutron was
established and in which its properties were first elucidated very soon
indicated that neutrons are capable of inducing nuclear transmutations.
Some of these early investigations have been discussed in Section 1. In
these the neutroninduced transmutations were observed in cloud cham
bers. Although the cloud chamber is a very effective instrument for
observing nuclear transmutations, its use imposes serious restrictions on
the types of nuclear reactions which can possibly be detected: The tar
get nuclei must be capable of being introduced, as an appreciable con
stituent, into the cloud chamber; the transmutation products must be
charged and have sufficient energy (range) to leave observable and
identifiable tracks. With relatively few exceptions, cloud chamber
techniques can only be applied to the study of nuclear transmutations
induced in light elements by fast neutrons.
With the announcement by Curie and Joliot (C41), in January 1934,
of the production of artificially radioactive nuclei, an entirely new field
of neutron investigation became available. 1 Their discovery prompted
1 The discovery by Curie and Joliot was made in experiments involving the bom
bardment of boron, magnesium, and aluminum by polonium alphaparticles. They
first reported the reaction
AF + a. > P 30 + n
p30 _> gi 30 + e +
and proved that the reaction product is an isotope of phosphorus, decaying by posi
tron emission, with a halflife of 3.25 min.
248 The Neutron [Pt. VII
Fermi to investigate the possibility of producing artificial radioactivity
by neutrons, and he soon announced (F13) that the bombardment of
aluminum and fluorine by neutrons does indeed induce such radioactiv
ity. Within a short time, Fermi and his coworkers at the University
of Rome succeeded in inducing artificial radioactivity in a variety of
elements through neutron bombardment (F12). In the next few years
the Rome group played a leading role in the investigation of neutron
induced nuclear reactions.
A significant step forward in the study of neutron reactions resulted
from the observation by Fermi, Amaldi, Pontecorvo, Rasetti, and
Segre (F14) that the neutroninduced radioactivity in silver can be sig
nificantly increased by interposing, between the source of fast neutrons
and the silver detector, a slab of paraffin. This effect was correctly
ascribed to the slowing down of the neutrons by collisions with hydrogen
nuclei, and it indicated that the probability of neutron capture increases
with decreasing neutron energy. This conjecture was confirmed by the
observation, by Moon and Tillman (M35), of an effect of the tempera
ture of the moderator on the neutron capture. The slowing down of
neutrons in hydrogencontaining materials will be discussed subse
quently. For the purpose of this summary it is, however, important to
note that the recognition of the strong dependence of neutron capture
probability on the neutron velocity was a significant step toward the
understanding of neutron reactions and that, for many years, sources
of neutrons slowed down in paraffin were the main tool for the investi
gation of neutroninduced nuclear reactions.
The fact that radioactivity can be induced by neutrons in the heaviest
elements is already a strong indication that the reactions responsible for
their production probably do not involve the emission of charged parti
cles since, for heavy nuclei, the Gamow barrier is a very strong deterrent
to charged particle emission. Furthermore, in many cases the radioac
tive nuclei were shown to be isotopes of the bombarded nuclei (F12).
In particular, the ~15 hr halflife /3 — activity resulting from neutron
capture by Na 23 , discovered by Bjerge and Westcott (B32), was identi
fied as an isotope of sodium by Amaldi, D Agostino, and Segre (A16) ;
the fact that the decay involves emission of p~ particles was interpreted
to imply that the radioactive isotope is Na 24 (rather than Na 22 ), indi
cating an (n,y) reaction. The observation that the probability of pro
ducing radioactivities in heavy nuclei increases with decreasing neutron
velocity represents an even stronger argument for the contention that
the reaction involved is that of radiative capture (n,y) — capture of a
neutron followed by gammaray emission. That this is indeed the case
was shown by Amaldi, DAgostino, Fermi, Pontecorvo, Rasetti, and
Sec. 2 A] Interaction with Nuclei 249
Segre (A19), who proved that the capture of slow neutrons by cobalt,
cadmium, chlorine, iridium, silver, and mercury is accompanied by
gammaray emission. 1 For slow neutrons, radiative capture is usually
the most important reaction. However, in the case of the capture of
slow neutrons by light nuclei, charged particle emission may be much
more probable. Thus, Chadwick and Goldhaber (C9) and, independ
ently, Amaldi, D'Agostino, Fermi, Pontecorvo, Rasetti, arid Segre
(A17) showed that the strong capture of slow neutrons by lithium and
boron is due to an (n,a) reaction. Somewhat later Burcham and Gold
haber (B75) demonstrated that the capture of slow neutrons by nitrogen
is due to an (n,p) reaction.
Perhaps the most important advance in the study of neutron reac
tions was the discovery of slowneutron resonances — the preferential
capture of neutrons of specific energies by certain nuclei. The first indi
cation of resonance effects was obtained in the experiments of Bjerge
and Westcott (B33), soon followed by the work of Moon and Tillman
(M35, T9), which established the resonance nature of some slowneutron
interactions. There then followed a period of rapid development of tech
niques for studying the properties of slowneutron resonances. Among
the most important of these were: the use of cadmium difference and
self indication methods by Fermi and Amaldi (F16) and by Szilard (S64) ;
the measurement of neutron slowing down lengths in paraffin using, as a
detector, the resonance in question, by Amaldi and Fermi (A20); the
use of boron absorption techniques for the measurement of resonance
energies by Frisch and Placzek (F45) and by Weeks, Livingston, and
Bethe (Wll) ; the development of the first slowneutron velocity selector
by Dunning, Pegram, Fink, Mitchell, and Segre (D20). These tech
niques, their development and extension, will be discussed in detail in
Section 3. Among the most significant of the early studies was the
proof, by Dunning, Pegram, Fink, and Mitchell (D19), that the very
large interaction of thermal neutrons with cadmium cannot be due to
neutron scattering and must, therefore, be assumed to result from the
radiative capture process.
The study of slowneutron resonance capture received great impetus
from the theoretical considerations of Bohr (B44) on the role of the com
pound nucleus in nuclear reactions. On the basis of Bohr's arguments,
it became possible to understand the existence of slowneutron capture
resonances, and to interpret their observed properties in terms of the
BreitWigner formula (B60) (previously derived on the basis of very
1 The emission of gammaradiation in the capture of neutrons by hydrogen had
previously been observed by Lea (L10).
250 The Neutron [Pt. VII
general considerations) for the energy dependence of the nuclear cross
section in the neighborhood of a resonance.
In addition to the work mentioned above, some of the investigations
which were important in confirming the theoretical ideas concerning
neutron resonances include those of Preiswerk and Halban (P30),
Frisch (F46), and Goldsmith and Rasetti (G19). The experiments and
theory of slowneutron resonances, up to 1937, are summarized by Bethe
(B24), by Bohr and Kalckar (B45), and by Moon (M36). The theory
and observation of slowneutron resonances, up to 1940, are discussed
by Peierls (P10).
While the most important advances in neutron physics were made in
the study of the properties of slow neutrons, the investigation of fast
neutron reactions was not completely neglected. Of the artificial radio
activities induced, by neutrons from RnBe sources, in light elements, the
pioneering work of Fermi and coworkers (F12) showed, by chemical
separation of the radioactive nuclei, that the responsible reactions were
mainly (n,p) and (n,a). In the case of the heavy elements, the radio
active products appeared, in most cases, to be isotopes of the target
nuclei. Fermi and coworkers advanced two alternative reaction possi
bilities: radiative capture, or the ejection of a neutron from the target
nucleus by a neutronneutron collision, i.e., the (n,2n) reaction. The
dependence of the probability of most neutron reactions on the neutron
velocity, discussed above, led to the assumption of the first hypothesis,
confirmed by the observations on sodium (B32, A16) and other ele
ments (A19). However, in 1936, Heyn (H56) proved that, for neutrons
of sufficiently high energy, the (w,2n) reaction could also be induced.
In the meanwhile, Dunning and coworkers at Columbia University
(D18, D19) were studying the reactions of fast neutrons with nuclei,
using RaBe sources. In addition, with the discovery, by Szilard and
Chalmers (S63), of the photodisintegration of beryllium by the gamma
rays from radium and its products, there became available neutron
sources in the intermediateenergy range. These were exploited by a
number of investigators (M22, L13, G25) for the study of neutron reac
tions in the 0.1 to 1 Mev range. Of primary importance in the study of
fastneutron reactions was the discovery, in 1933, of the possibility of
producing strong fastneutron sources by the use of particle accelera
tors through (d,n) reactions (Crane, Lauritsen, and Soltan, C30), and
(p,n) reactions (Crane and Lauritsen, C31). Of special significance in
this respect is the d + d reaction (Oliphant, Harteck, and Rutherford,
()3), which can be used with relatively lowvoltage accelerators.
Among the early results of fastneutron studies was the observation
by Lea (Lll) of the excitation of gammarays by the passage of fast
Sec. 2B] Interaction with Nuclei 251
neutrons through matter. Lea showed that the production of gamma
rays increased with the atomic number of the material traversed, and
that the observed effects were not due to a neutron capture process;
he ascribed them to the excitation of nuclei in the inelastic scattering of
fast neutrons, followed by nuclear deexcitation through gammaray
emission. Similar results were obtained by Kikuchi, Aoki, and Husimi
(K12). It was shown, by Danysz, Rotblat, Wertenstein, and Zyw
(D3), by Ehrenberg (E2), and by Collie and Griffiths (C20), that fast
neutrons are indeed slowed down, far in excess of what could be ac
counted for by elastic scattering, in their passage through heavy ele
ments.
The last of the significant new neutron reactions to be uncovered
was neutron fission, discovered by Hahn and Strassman (H4) early in
1939. This discovery was the result of a long series of investigations
which attempted to understand and interpret the results of the neutron
bombardment of uranium, first reported by Fermi et al. (F12). Imme
diately after the announcement by Hahn and Strassman that isotopes
of barium, lanthanum, and cerium are certainly among the products of
the capture of slow neutrons by uranium, Meitner and Frisch (M23)
showed that a fission reaction (division into two fragments of roughly
equal mass) was to be expected from energetic considerations; such a
reaction, they observed, is highly exoergic. Physical evidence for the
expulsion of highspeed nuclear fragments in the slowneutron fission of
uranium was soon obtained by Frisch (F49) and by Joliot (Jll). The
history of the discovery and early investigation of the fission reaction
has been reviewed by Turner (T17).
B. General Considerations
Given a complete knowledge of the law of force between nucleons and
of the law of motion (quantum mechanics) which governs their mutual
interaction, it is conceivably possible to set up a program of computing
the static properties of all nuclei and the detailed characteristics of the
interactions between neutrons and nuclei. Actually, this program is
still in its preliminary stages, if indeed the goal is at all attainable for
any but the lightest nuclei. In the first place, the law or laws of force
are not yet sufficiently well established. In the second place, their appli
cation to complex, heavy nuclei would involve the solution of the equa
tions of motion for the manybody problem, in which the components
are closely spaced and strongly interacting (so that the approximation
methods, which are so useful in problems involving the electrons in an
atom, may not be applicable to nuclei). Finally, it is by no means cer
tain that specifically manybody forces (not detectable in or predictable
252 The Neutron [Pt. VII
from a study of the nucleonnucleon interaction) do not come into play
in the interaction of the nucleons in a nucleus.
Nevertheless, a number of attempts have been made to understand
the properties of nuclear (especially neutron) interactions in terms of
approximations in which the nucleons in the nucleus are assumed to
behave as moreorless independent particles (A19, Bll, B21, Pll).
The predictions of such models were soon found to be at strong variance
with experimental observations, especially with regard to the properties
of slowneutron resonances; for, while these models predict strong ther
mal neutron capture cross sections and slowneutron resonances, the
widths of the levels and the spacing of levels, as predicted, were far in
excess of those observed. Furthermore — and most significant — the
independent particle models were unable to account for the predomi
nance of radiative capture over scattering in most slowneutron reso
nances.
To overcome these difficulties Breit and Wigner (B60) introduced the
idea that the slowneutron capture process must involve more than one
of the nuclear constituents; their arguments were based mainly on an
analogy with certain atomic and molecular phenomena. The conceptual
and theoretical groundwork for the understanding of nuclear reactions
was laid by Bohr (B44) in a work of classic and farreaching significance.
Bohr emphasized the necessity of going to the opposite extreme from the
independent particle picture, and of recognizing that the nucleons in
the nucleus interact very strongly with each other. Thus, a neutron,
on entering the nucleus, very rapidly loses its identity as bombarding
particle, sharing its energy among all the constituents of the nucleus.
Accordingly, a nuclear reaction must be regarded as taking place in two
distinct and separable stages. In the first, the incident particle is in
corporated into a compound nucleus, sharing its energy — kinetic and
binding — with the rest of the nucleus. In the second stage, the com
pound nucleus gives up its excitation energy by any one of the possible
means at its disposal. These means include radiation, particle emission,
and neutron reemission. In this competition among the various (ener
getically) possible deexcitation processes, radiation can compete very
favorably with particle emission, especially in heavy nuclei, since parti
cle emission (including neutron reemission) requires the concentration
of a large fraction of the excitation energy into one particle, a process
which is relatively improbable. Furthermore, because of the sharing of
the excitation energy among the nuclear constituents, the compound
nucleus has a relatively long lifetime, which results in comparatively
sharp resonances. These ideas lend themselves to quantitative de
velopment in terms of a statistical theory of nuclear energy levels and
Sec. 2B] Interaction with Nuclei 253
of nuclear reactions (B45, B23, F43, W16, L2) ; they provide the basis
for many theoretical investigations of the properties of slowneutron
resonances (B25, K2, S37, B61, W29, F27, F28, A6). In their extreme
form, they suggest the liquid drop model of heavy nuclei, which has been
so useful in understanding the dynamics of the fission process (B46).
So successful were the ideas of Bohr, and their quantitative develop
ment, in understanding, correlating, and predicting the properties of
nuclear reactions, as well as some of the general features of the stable
nuclei (binding energies, curve of stability) that the independent parti
cle model was completely discredited. There remained, however, cer
tain features of nuclear behavior which, although they could be fitted
into the general framework of the statistical model, required concessions
in the direction of the independent particle model; thus, it was neces
sary to treat somewhat differently nuclei with odd and with even num
bers of nucleons. Furthermore, the success of the independent particle
model, as applied by Schmidt (S5), in correlating nuclear magnetic
moments indicated that the ground states, at least, of nuclei require
such a treatment.
The accumulation of nuclear data has recently led to the recognition
by Mayer (M7) that many nuclear phenomena indicate the existence of
a nuclear shell structure, first predicted by Elsasser (E4), thereby re
viving the independent particle model. The particular stability of
nuclei containing 2, 8, 20, 28, 50, 82, or 126 neutrons or protons is borne
out by a large number of nuclear data. A number of forms of the inde
pendent particle model have been invoked to derive the above "magic
numbers" (F6, N2), of which the most successful has involved the
assumption of a strong spinorbit coupling, by Haxel, Jensen, and
Suess (H49) and by Mayer (M8). Table 3 (after Mayer, M9) partially
summarizes the predictions of this model.
While the statistical model and the independent particle model repre
sent opposite extremes, it appears necessary nevertheless to recognize
that both points of view are applicable, each in its domain, and that the
two approaches are complementary. Thus, the independent particle
picture appears to be required for the understanding of the ground and
lowlying excited states of nuclei, whereas the statistical model is more
appropriate for phenomena involving relatively large nuclear excitation,
as is the case in almost all neutron reactions. Weisskopf (W17) has
pointed out that the nuclear situation seems analogous to that of the
electrons in a metal, where the electrons can be described in terms of an
independent particle model for the states of small excitation because of
the effects of the Pauli principle, despite the fact that they are inter
acting very strongly with each other.
254
The Neutron
TABLE 3
[Pt. VII
Order of Energy Levels Obtained from Those of a Square Well
Potential by SpinOrbit Coupling
(After
Mayer, M9;
Oscillator
Square
Well
Level
Spin
Number
Number
Total
Number
Term
of States
in Shell
Number
Is
lSi A
2
2
2
1
lp
ipy*
4
lPH
2
6
8
2
Id
ldtft
6
(14)
2s
&X
2
ld%
4
12
20
3
1/
V»
8
8
28
2p
2PK
4
6
2PK
2
(12)
(40)
4
1(7
ig%
10
22
50
2d
2d H
^g%
2d H
3sy 2
6
8
4
2
5
Ih
lhi$i
12
32
82
2/
2/54
8
3p
3pk
3ph
4
6
2
10
6
3d
4s
2?«
14
44
126
Sec. 2B] Interaction with Nuclei 255
Nevertheless, it is necessary to observe that neither model provides a
complete description of nuclear phenomena in any energy region; the
strong interaction undoubtedly influences the properties of the low
lying nuclear levels, while (as we shall have many occasions to observe)
the shell structure has important effects on the characteristics of neutron
reactions, even at relatively high levels of excitation.
The properties of nuclear reactions have been discussed in Part VI.
In this section we shall review these properties as they apply to neutron
reactions, and summarize the status of the present knowledge of the
interactions of neutrons with nuclei. The available data on neutron
cross sections have been compiled by Goldsmith, Ibser, and Feld (G20)
(October 1947), by Adair (A2) (July 1950), and by an AEC Committee
(A0) (1952). The discussion here will lean heavily on these compila
tions, in which references are given to the original investigations.
1. Energetics of Neutron Reactions. When a neutron of kinetic
energy E is captured by the target nucleus, the product (compound)
nucleus acquires an excitation of energy E' = E + e. e is the binding
energy of the neutron in the product nucleus. This excitation makes
possible a variety of nuclear reactions, depending on the particle or par
ticles whose emission from the compound nucleus requires less than the
energy E'. The characteristics of the reactions observed depend on the
properties of the compound nucleus at the excitation energy E', and on
the competition between the possible modes of deexcitation of the
compound nucleus.
Clearly, for neutrons of kinetic energy E <SC e, the characteristics of
the observed reactions will be critically dependent on the value of e.
The variation of e with Z and A may be summarized as follows : For
nuclei of A < 20, e exhibits large (periodic) fluctuations from nucleus
to nucleus. For nuclei of A "> 20, the values of e, on the average, in
crease slowly from ~8 Mev to ~8.5 Mev at A — 130, and then decrease
slowly to <~7.5 Mev for the heaviest nuclei. However, these values
apply to the stable nuclei; the capture of a neutron by a stable nucleus
usually results in a radioactive product nucleus for which the neutron
binding energies are somewhat smaller than for the stable nuclei. Thus,
the values of e, for captured neutrons, average ~78 Mev for A be
tween 20 and 150, falling slowly to ~6 Mev for neutrons captured by
the heaviest nuclei.
However, neutron binding energies fluctuate significantly from nucleus
to nucleus. The capture of a neutron by a nucleus with an odd number
of neutrons results in a greater (by ~l2 Mev) release of energy than
capture by adjacent nuclei of even neutron number. Quite large devia
tions from the average may also occur in the vicinity of neutron magic
256 The Neutron [Pt. VII
numbers; thus, the value of e for a neutron captured in a nucleus lack
ing but one neutron for a closed shell will be anomalously high, while «
for a neutron captured by a nucleus already having a completed neutron
shell is anomalously low. Furthermore, such magic number effects
appear to persist for neutron numbers ranging considerably to either
side of a magic number (K23).
In those (relatively rare) cases in which the masses of the target and
product nuclei are known, the neutron binding energy can be computed :
e = 931.16 {M(X A ) + M(n)  M(X A+1 )} in Mev (16)
For light nuclei the masses of the nuclei involved are, in general, avail"
able and the above computation can be made. For nuclei of A "> 30
the masses are usually not known, and it is then necessary to resort to
the general considerations described above, if it is desired to estimate
the value of «.
From the above expression for e, it may be seen that it is not necessary
to know the absolute value of the nuclear masses in order to compute e,
but only the mass difference, M(X A+1 ) — M(X A ). Nuclear mass dif
ferences are frequently known as a result of measurements of the energies
of reactions involving the two nuclei in question, even when the absolute
values of the masses are unknown. Thus, in the region of the naturally
radioactive nuclei, neutron binding energies can be computed with the
aid of the observed energies of alpha and betaemission (W9).
For a few nuclei the energy release accompanying slowneutron cap
ture has been measured directly by observation of the energy of the
gammaradiation emitted when the excited compound nucleus decays to
its ground state. In this fashion, Bell and Elliott measured the binding
energy of the deuteron (B16). Unfortunately, for nuclei more compli
cated than the deuteron the radiative deexcitation of the compound
nucleus is usually achieved by the emission, in cascade, of a number of
gammarays. Since there are usually a large number of energy levels
of the compound nucleus available to such cascade gammaray emis
sion, the capture gammaray spectrum is quite complicated. However,
in certain favorable cases the emission of a single gammaray, carrying
away all the excitation energy, occurs in a reasonably large fraction of
the decays; the energy of this gammaray gives, directly, the value of e.
Such direct transitions to the nuclear ground states have been observed
in a number of nuclei by Kinsey, Bartholomew, and Walker (K15).
In general, the nuclear reactions in which we are interested can be
represented symbolically as
on 1 + Z X A > ( Z X A+1 )* * z x a + z _ z Y A+l  a + Q (17)
Sec. 2B] Interaction with Nuclei 257
[For many light nucleus reactions the intermediate (compound) nucleus
step should be omitted, since it has no independent existence.] The re
action is frequently abbreviated as X. A (n,x)Y A+1 ~ a , or simply as
~K. A (n,x). The Q value of the reaction is determined by the masses of
the nuclei involved :
Q = 931.16{AT(n) + M(X)  M(x)  M(Y)} in Mev (18)
or, conversely, a measurement of the Q value can be used to determine
one of the masses, if the other three are known, or the mass difference
M (X) — M (Y), from the masses of the neutron and the ejected particle.
A positive Q value indicates an exoergic reaction, i.e., a reaction that
can take place (from the point of view of available energy) with neutrons
of zero kinetic energy; a negative Q value means that the reaction is
endoergic and is not possible for neutrons of kinetic energy less than a
certain threshold value:
f[M(X) + M(n)]) (04 + 1)1
The factor in brackets arises from the conservation of momentum, which
requires that some of the neutron kinetic energy be expended in pro
viding for the centerofmass motion of the system. The possibility
that a given reaction can be initiated by thermal neutrons depends on
having Q > 0. Two reactions are, however, always possible for thermal
neutrons: (1) elastic scattering (or reemission) of a neutron (n,n), for
which Q = 0; (2) radiative capture (n,y), for which Q = e.
2. Theory of Neutron Reactions ; the Compound Nucleus. According
to the Bohr model, most neutron reactions can be regarded as taking
place in two distinct stages — the formation of an excited compound nu
cleus, and its subsequent decay. In our survey of the theoretical conse
quences of these ideas, we lean heavily on the schematic theory of nuclear
reactions developed by Weisskopf and coworkers (F27, F28, W17).
This theory is based on some general assumptions regarding the struc
ture of nuclei: that the nucleus has a welldefined surface, which is a
sphere of radius R, outside of which the interaction with neutrons is
negligible; that, once inside the nucleus, the neutron interacts very
strongly with the nuclear constituents, rapidly sharing its excitation
energy among them.
Because of the saturation property of nuclear forces, the average
distance between nuclear constituents is independent of their number,
258 The Neutron [Pt. VII
A. Thus, the volume V of a nucleus is proportional to A, and its radius
can be approximated by the relationship
R = r A 1A (20)
with r between 1.3 and 1.5 X 10~ 13 cm.
A neutron of kinetic energy E will, on penetrating the nuclear sur
face, suddenly find itself in a region where its kinetic energy is E' ^ E
+ e, where e is the average kinetic energy of a nucleon inside the nu
cleus. Such a sudden change of kinetic energy results, from wave
mechanical considerations, in a reflection; the probability for penetra
tion of the surface is given by
4JcK
P = ttt^, (2D
where
(k + KY
„ 2ME 1
k 2 = — — =  (22a)
X
2
for the incident neutron, and
K2 = ™<l+A = k2 + K j ,., w.
fi 2 h 2
for the neutron inside the nucleus. (K and k are the wave numbers
associated with the neutron inside and outside the nucleus, respectively.)
The value of K can be computed on the assumption that the nucleons
in the nucleus obey FermiDirac statistics, distributing themselves
among the various possible states of momentum, spin, and isotopic spin
(charge) so that no state is occupied by more than one nucleon, giving
£ = ( ) ( — ) ^ 20 Mev
V 2M I \16ttVJ
(23)
K ^ 1.0 X 10 13 cm 1
for r = 1.5 X 10 13 cm. For slow neutrons, k « K ^ K , and
4fc
P ^ — « 1 (24)
Ko
Despite the smallness of P for slow neutrons, the probability of neutron
capture to form a compound nucleus may be quite large since slow
neutrons have a large extent (wavelength). The capture probability
depends critically on the proximity of the excitation energy (E' =
E + e) to one of the energy levels of the compound nucleus, being
essentially a resonance phenomenon. We postpone the discussion of
Sec. 2B] Interaction with Nuclei 259
the formation of the compound nucleus to the next section and confine
ourselves, at this point, to consideration of what happens after the com
pound nucleus has been formed.
Once a neutron has entered the target nucleus, it very rapidly shares
its energy among the nuclear constituents, forming a compound nucleus
in an excited state. Furthermore, owing to the smallness of P (which
works both ways), the neutron has a very small probability of reemerg
ing, even if it should find itself in the relatively improbable circumstance
of possessing all the excitation energy. Consequently, the lifetime of
the compound nucleus is very long compared to the time required for
its traversal by a nucleon. In fact, for slow neutrons captured in nuclei
containing a relatively large number of constituents (say A >^ 100) the
most favorable mechanism for deexcitation of the compound nucleus is
usually the emission of gammaradiation, which is itself a relatively
improbable process.
These considerations can be made somewhat more quantitative. The
compound nucleus, formed in an excited state, may have a number of
(energetically) possible modes of decay which always include radiation
(n,y), neutron reemission (n,ri), and may also include the emission of
one or more particles (n,a), (n,b), etc. These compete freely with each
other, each mode of decay being characterized by a meanlife, t, (the
lifetime which the excited state would have if all other possible modes of
decay were turned off). The meanlife of the excited state
1 ^ 1
=£ (25)
T ,■ Ti
is associated, according to the uncertainty principle, with a finite width
(energy spread) of the excited state of the compound nucleus:
h
r =  (26)
T
Correspondingly, we can define a partial width for the z'th mode of
decay:
h
r, =  (27)
so that
r = Er f (28)
i
The characteristics of the compound nucleus decay are determined by
the relative values of the partial widths for the various possible modes
of decay. For the case of gammaray emission, the entire excitation
260 The Neutron [Pt. VII
energy is available to the gammaradiation. Since, for slow neutrons,
e » E and E' « e , the probability of gammaray emission is essentially
independent of the energy of the incident neutron. However, T y «
constant may be expected to depend to a certain extent on the angular
momentum properties of the state involved, so that the constant will
not be the same for all the states of the compound nucleus.
For particle emission, however, the situation is quite different. In
this case, in addition to kinetic energy, the emerging particle must also
be supplied with its binding energy. Since the excitation energy is
shared among the many particles in the compound nucleus, the possi
bility of emission of a particle depends on the concentration of sufficient
energy into this particle to allow it to escape. Let T a be the average
time between such rearrangements of the nuclear constituents as would
permit the emission of particle a. The frequency of emission of the
particle a (l/r a ) is then given by the product of the frequency of such
favorable configurations (1/T a ) and the probability that the particle a,
given the requisite amount of energy, can penetrate through the nuclear
surface :
1 Pa
 = V (29)
Ta ia
Hence
h hP„
r a =  = — (30)
T a ±a
According to the previous considerations, and assuming that the kinetic
energy of the emerging particle is small compared to its energy in the
nucleus, and that it has zero angular momentum,
4fc„
P a = — Qa (31)
fro
where G a is the Gamow barrier for particle a; G n — 1 for neutrons.
Thus
4hG
r„ ^ — — K = 7a G a E a * (32)
The above relationship applies only to the emission of particles of
zero angular momentum. It involves the kinetic energy of the emerg
ing particle, E a , through the factor E a y * and through the energy de
pendence of the Gamow factor, (?„, when the particle is charged. It
involves the properties of the compound nucleus through the factor K 0)
which is essentially independent of the nucleus involved, and the perio
dicity T a . The last factor depends quite critically on the specific nu
Sec. 2B] Interaction with Nuclei 261
cleus involved and on the excitation energy of the compound nucleus
state; this dependence on the properties of the compound nucleus is
through the nuclear level spacing, at the excitation energy involved, as
indicated by the following simplified considerations, due to Weisskopf
(W17): Assume that the energy levels of the compound nucleus are
equally spaced,
E n = E + nD (33)
Owing to the strong interaction, the nuclear wave function is given by
a linear combination:
£, iE n t iE t * inDt
* = 2, a n<l>n exp —  — = exp —  — 2, a n<t>n exp —  — (34)
n=0 U B n=0 h
The periodicity T, which is defined as the time required for the nucleus
to return to a previous configuration, is given by the condition
 *(t + 2xh/D)  =  ¥(t + T)  =  ¥(*)  (35)
whence
2irh
T = — (36)
Although the level spacing in the compound nucleus is far from uni
form, the above relationship nevertheless holds approximately for our
case, where D a is the average level spacing (of the levels which are
capable of decaying through emission of the particle a in a state of
given angular momentum) in the energy region of the excitation energy.
Combining Eqs. (32) and (36),
2G a D a
r« « — k a = CD a G a E a 1A (37)
for the emission of particles of zero angular momentum. The above is
a much simplified derivation of a wellknown relationship between level
spacing and level width (B21, B46, F27), which is borne out by the
observed neutron widths for slow neutrons (W31, B34).
The E y% dependence of the particle width on its kinetic energy also
follows from very general considerations involving the density of mo
mentum states available to the outgoing particle, in the case of emis
sion of zero angular momentum particles. For the emission of particles
of higher angular momentum, I, there is an additional energy depend
ence, which can be expressed as follows:
r„; = (21 + l)CD al G a iE a ^v al (38)
262 The Neutron [Pt. VII
where v a i = vi(k a R), R being the nuclear radius. Expressions for vi{x)
are (F27, B34)
f o = 1
(39)
x 2
v l
(1 + x 2 )
X 4
V%
(9 + 3x 2
+
* 4 )
v 3 =
(225 + 45z 2 + 6x 4 + a; 6 )
These have the limiting values
y2l
vi — > 1 for x — > oo
Thus, for the emission of lowenergy particles (k a R « 1), we need con
cern ourselves only with I = 0. At sufficiently higher energies, states
of i! > come into play.
Because of the selection rules associated with angular momentum
changes, the absorption and emission of particles of different angular
momenta usually involve different energy levels of the compound nu
cleus. For a given energy level of the compound nucleus, the emission
of a given particle in different states of angular momentum can usually
be treated on the same basis as the competition for the emission of dif
ferent particles.
So far, the spacing of the levels in the compound nucleus, D, has en
tered our considerations only as a parameter in the level width formulas.
Clearly, the level spacing is also of primary importance in the formation
of the compound nucleus, since the probability of neutron absorption
depends on the proximity of the nuclear excitation energy to one or
more of the levels of the compound nucleus.
On the basis of the Bohr model, it is possible to obtain a rough, semi
quantitative picture of the dependence of the level spacing of the nu
cleus on the excitation energy. For low excitation energies (of a few
Mev or less) the possible modes of excitation of the compound nucleus
are few, and the energy levels are widely spaced. The excitation of the
lowlying levels will involve only one or a few nucleons, and the details
of the lowlying levels will depend on the nucleus involved, being
Sec. 2B] Interaction with Nuclei 263
strongly influenced by such phenomena as magic numbers. As the en
ergy of excitation is increased, it becomes possible to excite more modes,
involving a greater number of nuclear constituents, and the levels be
come more closely spaced. For large excitation energies (say ~10 Mev)
an expression for the spacing of energy levels may be obtained from
thermodynamic considerations, and is of the form
w(E') = Cexp2(aE') 1A (41)
w(E') = 1/D is the level density (number of levels per unit energy inter
val) at the excitation energy E'. Typical values of C and a, for a number
of odd values of A, are given in Table 4; the level densities for even A
nuclei appear to be somewhat lower.
TABLE 4
Constants Determining the Level Spacings of Some Odd A Nuclei,
Eq. (41)
(Derived by Blatt and Weisskopf, B34, from available data)
A C (Mev" 1 ) a (Mev" 1 )
27
0.5
0.45
55
0.33
1.3
115
0.02
6.5
201
0.005
12
Figure 4a is a schematic energy level diagram for two nuclei, A ~ 55
and A = 201. Aside from the details, which depend on the nucleus in
volved, the general characteristics of the level scheme for the heavier
nucleus can be derived from that of the lighter by a contraction of the
energy scale. This feature is qualitatively expected from the Bohr
model since the heavier the nucleus, the more particles are available to
participate in the excitation, and the greater the level density at a given
excitation energy. Figure 4b shows the same level diagrams greatly
magnified in the region of excitation corresponding to the absorption of
a neutron.
As implied by Fig. 4b, the level spacing of interest, in considering
reactions involving neutron capture, is that which corresponds to the
excitation E' — E + « of the compound nucleus. At first glance it
might be expected that, for neutrons of moderate energy, this would be
primarily determined by the binding energy, e. However, evidence has
recently been presented by Harris, Muehlhause, and Thomas (H40) and
by Hurwitz and Bethe (H81) that this is not the case. Thus, Harris
et al. have presented data which indicate that the level densities at E'
264
The Neutron
[Pt. VII
10
P 5
AZ55
e~
A = 201
Fig. 4a. Schematic energy level diagrams for two nuclei, A =* 55 and A « 201.
The average level spacings are according to the statistical theory (Eq. 41), and the
constants are from Table 4. The details of the structure, i.e., the relative positions
of adjacent levels and the fluctuations in level spacing, are entirely fictitious, cor
responding to no known nucleus. They are intended to illustrate the possibility
of large deviations from the average.
Sec. 2B]
Interaction with Nuclei
265
« e, for capture of neutrons by odd protoneven neutron and by even
protonodd neutron nuclei, are essentially the same (for nuclei of roughly
the same A) although e is considerably smaller for the former than for
Mev
1.0
0.9
0.8
0.7
0.6
6q
II
&1
0.5
0.4
0.3
0.2
0.1
A ==55
AS 201
Magnification
X 10
X 20,000
500
400
300
200
100
Fig. 4b. Energy level diagrams of Fig. 4a greatly magnified in the region of the
neutron binding energy.
the latter (by ~1 Mev). They ascribe the difference in e to a lowering
of the groundstate energy of the compound nucleus of even Zeven N,
as compared to that of odd Zodd N; the level densities corresponding to
e are, however, the same for both types. Hurwitz and Bethe concur and
266 The Neutron [Pt. VII
present arguments for the hypothesis that the level density of the com
pound nucleus, at E' « e, is determined primarily by the binding energy
of the target nucleus (rather than by that of the compound nucleus);
target nuclei with high binding energy (e.g., eveneven or magic number
nuclei) result in compound nuclei with low level spacing. Data of
Hughes and coworkers * on the radiative capture of fast neutrons, how
ever, appear to be in disagreement with this point of view.
In any event, it should be pointed out that the effect of the apparent
validity of the independent particle model at low excitation energies will
be to invalidate the applicability of Eq. (41) to excitation energies of a
few Mev. Available evidence, mainly from inelastic scattering and (p,n)
reactions, indicates that level densities at low excitation energies are
rather smaller than predicted by Eq. (41), and that the exponential be
havior does not really set in until values of E' >, 35 Mev are achieved.
The constants of Table 4 should be regarded as applying, roughly, to
energies E = E' — a ~ 02 Mev.
The compound nucleus theory of nuclear reactions, outlined above,
will be expected to break down at very high excitation, for at high
enough energies the probability of escape of the incident neutron after
a single traversal may be quite large, i.e., the nucleus is more or less
transparent to highenergy neutrons. Thus, the incident neutron may
not have sufficient opportunity to share its energy among the nuclear
constituents before it (or one of the other nucleons) escapes. In this
respect, it is to be expected that the absorption of a charged particle by
a heavy nucleus will, even for high incident energies, be more likely to
lead to a true compound nucleus, since the escape of the charged particle
from the nucleus is impeded by the Gamow barrier.
3. Formation of the Compound Nucleus; Resonances and the Breit
Wigner Formula. In accordance with the twostage picture of neutron
reactions (formation of a compound nucleus; subsequent emission of a),
the (n,a) cross section is
<j{n,a) = <r c — (42)
The first factor, <r c , is the cross section for the absorption of the incident
neutron into an excited compound nucleus; the second factor represents
the relative probability for the deexcitation of the compound nucleus
through the emission of a.
The cross section for the formation of the compound nucleus depends
critically on the proximity of the excitation energy (E + e) to one or
1 Private communication.
Sec. 2B] Interaction with Nuclei 267
more of the energy levels of the compound nucleus. In the case where
only one level of the compound nucleus is involved (all other levels
being far enough removed from the value of the excitation energy so
that their influence on a c may be neglected), the energy dependence of
<r c is given by a characteristic resonance formula,
r„r
<r c = ttK*1 — ■
(E  i
whence
<r c = tX 2 / ~^r — (43)
J (E  E T f + r 2 /4
a{n,a) = irX 2 / ~ — (44)
K (E  E r ) 2 + T 2 /4
Equation (44) is the wellknown BreitWigner onelevel formula; the
symbols have been previously defined, except for E T , which is the neu
tron kinetic energy for which the excitation energy is equal to the dif
ference between the ground state and the energy of the compound nu
cleus level involved in the reaction, and /, a statistical weighting fac
tor which depends on the neutron spin, s = \ ; its orbital angular mo
mentum, lh; the total neutron angular momentum quantum number
j = I ± \ ; the spin of the target nucleus, J; and the spin of the com
pound nucleus level involved, J = I+j,I + j— 1, • • , \ I — j\:
(2/+1) _p/ + l)
(27+l)(2s+l) 2(27+1)
The factor / is simply a measure of the probability that the neutron angu
lar momentum, jh, and that of the target nucleus, lh, either or both
assumed oriented at random (unpolarized), will have a relative orienta
tion appropriate to the necessary total angular momentum, Jh, required
for the formation of the compound nucleus level involved in the reso
nance. For slow neutrons, I = 0, j = \, and J = I ±\; whence for
I ± 0,
1
1
1 ±
1
f^/ .
for large I (45a)
2L (27+l)J 2
and /o = 1 for 7 =
For neutrons of I > there are two possible "channel" spins, j = I
± , each of which combines with the target nucleus spin 7 to give the
possible compound nucleus spin values J. Some of these can be formed
through only one channel; for these, the statistical factor, fi(J), is un
ambiguously given by Eq. (45). Other J values can be formed through
both channels; for such levels the question arises as to whether the
proper value of the statistical factor is fo(J) or 2f t (J).
268 The Neutron [Pt. VII
Let us consider a specific example: Assuming a Presonance (1=1)
for a target nucleus of / = §, the possible J values of a compound nu
cleus state are
J = 2,1 fori = f
J = 1,0 forj=f
The statistical weights are, from Eq. (45),
/i(2) = f
/i(l) = f
/i(0) = i
For levels of J = 2 or J = 0, there is no ambiguity. The J = 1 state
illustrates the possible ambiguity. On the one hand, a given J = 1
level may be such that it interacts only with one of the two possible
channel spins (or one of two possible orthogonal linear combinations of
/ = J and j = ); under these circumstances, the statistical factor
/i(l) = i O n the other hand, the interaction might be completely
independent of j, depending only on the I value, which is the same for
both channels. In this case, /i(l) = 2 X f = f •
In either event, the ambiguity disappears if many levels of the com
pound nucleus are involved in a given interaction, since the sum of the
statistical factors, over all possible J and j values, is ^2fi(J) = 3. This
is a special instance of the general relationship J,i
T,fi(J) = (2l+l) (45b)
The situation with regard to the statistical factor is even more com
plicated if the same compound nucleus state can be formed by different
neutron orbital angular momenta (12). However, this added compli
cation is seldom of practical importance, since the neutron widths will
be very different for different values of I, which fact usually effectively
eliminates the influence of all but the lowest possible I value. In addi
tion, for a given compound nucleus state the parities of both the initial
and the final states are well defined. Hence, the choice of possible I
values is even further limited, adjacent values not being simultaneously
available.
According to Eqs. (43) and (44), neutron capture and the accompany
ing reactions are characterized by sharp resonances for incident neutron
Sec. 2B] Interaction with Nuclei
269
energies E = E r , with the maximum (peak) cross section
47rX 2 / z r ni
cole  r
(46)
4irX 2 fiT n iT a
<r i(n,a) =
(47)
and (full) width at half maximum
h
ae = r = 
T
(48)
Equation (44) applies to the case of elastic scattering as well as to the
case of emission of some other particle. However, in addition to the
scattering which involves the formation of a compound nucleus (so
called capture scattering) there is a second type of elastic scattering
process known as potential scattering. Potential scattering results from
the small penetrability of the nucleus to slow neutrons (Eq. 24) and can
usually be described in terms of the scattering of an impenetrable sphere,
for which the scattering cross section would be
op = ttX 2 £ (21 + 1) e**  1  2 = 4*X 2 £ (21 + 1) sin 2 & (49)
i i
j is the phase shift of the scattered partial wave of orbital angular
momentum Zh. For slow incident neutrons, R <5C X = 1/fc and
So = kR & >0 « So (50)
whence
<r p ^ a p0 = 4xX 2 sin 2 kR ^ 4irR 2 (49a)
The phase shifts corresponding to scattering by a rigid sphere can be
obtained from the following general expression (F27, L23), with kR = x
where J and N are, respectively, the Bessel and Neumann functions.
So = x
x _
Si = x h cot x
_. x 2  3
£ 2 = x — x + cot ■
Sx
3x x(x 2  15)
£q = a: H cot —
?3 2 fa 8  15
(50b)
270 The Neutron [Pt. VII
For x « 1,
So = x
x 3 x 5 x 7
/x b x 1 \
\7 21/
)
(50c)
* 3 ~225\7 ~ 7'
A more useful approximation is, in the limit x <JC  21 — 1 ,
x 2l+l
& « tan 1 (50d)
At the other extreme, for x 5S> I 2 ,
It
6 « a:   (60e)
Values of the phase shifts vs. x have been tabulated by Lowan, Morse,
Feshbach, and Lax (L23).
Equation (49) applies to the general case of scattering by an arbitrary
potential (as well as to scattering by a rigid sphere) :
<r,c = 4ttX 2 J2 W + 1) sm 2 ft (49')
i
The phase shifts, ft, are obtained from the asymptotic behavior of the
radial functions, u h in terms of which the wave function of the system is
written :
ii>(r,0) = 22 Ui(r)Pi(cos 0) (49")
i
The functions ui satisfy the radial Schrodinger equations:
«i" +
;?>— ^
ui = (49"a)
For slow and intermediate neutrons, X ^> R and only I = scattering
need be considered; i.e., ft> <3C /3 , and c sc = 4xX 2 sin 2 /3 .
The value of ft, depends, of course, on the form of the scattering po
tential V (in Eq. 49"a, with 1 = 0). A number of possibilities are illus
trated in Fig. 5 in which u is plotted in a number of hypothetical cases.
Sec. 2B] Interaction with Nuclei
u (r)
271
Fig. 5. Illustrative examples of the joining of the neutron wave functions at the
nuclear boundary, leading to different phase shifts and scattering lengths. Case
(a) applies far from a resonance, on the assumption of a "rigid sphere" nuclear
model. Cases (b), (c), and (d) represent resonance scattering. Case (e) is intended
to illustrate scattering by a finite potential well.
272 The Neutron [Pt. VII
It is assumed that the potential V acts only in the range — R; foj
r > R, u (r > R) = sin (kr + p ). The intercept a of u (r > R) on th<
r axis, indicated with an arrow in each case, determines /3 = — a/%.
a is called the scattering length or scattering amplitude; according tc
convention, positive a will be assumed to mean that the intercept is on
the right — cases (a), (d), and (e) — i.e., a negative phase shift, /Jo
Figure 5a illustrates the case of scattering far from a resonance; it is
seen that a ^ R, /3 ^ R/X « 1, and <r sc ^ AtvR 2 . (Figure 5 greatly
exaggerates the magnitude of R; for slow neutrons, R, the nuclear
radius, is ~10~ 12 cm and X is ~10 8 cm.) This case is essentially the
same as scattering by a rigid sphere, for which u (r < R) = 0.
Cases (b), (c), and (d) represent various stages of scattering near and
at resonance. It is seen that the resonance condition (case c) corre
sponds to a ^ ±X/4, ft, ^ ±x/2, and <r ^ 4ttX 2 .
Case (e) is supposed to represent the class of scatterings by a finite
potential well, rather than by a rigid sphere (e.g., neutronproton scat
tering). Under such circumstances, the scattering length depends in
detail on the depth and range of the potential. In particular, if we con
sider scattering by a potential of fixed depth but continuously increasing
range, the value of a will go through "resonances" (involving both large
magnitude and change in sign) at certain values of R. Such "nuclear
size resonances" have recently been invoked by a number of authors
(F38, M40, S33) to explain large variations in the potential scattering
of adjacent nuclei.
For neutron energies far from a resonance, potential scattering is the
most important scattering process. In the region of a resonance, cap
ture scattering predominates. However, for the scattering of a neutron
of given I, the capture and potential scattering are coherent and inter
fere with each other. Thus, the BreitWigner formula must, in the case
of elastic scattering, be modified to
a(n,n) = 7rX 2 /j
i?ni
e 2i c — 1
2
+ <r/ (51)
(E  E r ) + iT/2
where <r p ' is the incoherent part of the potential scattering, 1
a p ' = a p  tX% e 2iit  1  2 (49b)
For most observed neutron resonances, the neutron kinetic energy is
sufficiently low so that we have only to concern ourselves with I =
interactions. Furthermore, except for the light nuclei (A ^ 25), the
1 Note that the usual coefficient (21 + 1) is contained in the statistical factor /;.
Sec. 2B] Interaction with Nuclei 273
only reactions of significance in the region of observed resonances are
(n,ri) and (n,y). For such resonances (with kR <<C 1),
r = r» + r r
<r(n,y) = 7rX 2 / 
L n* y
(E
 E r ) 2 + T 2 /4:
<r(n,ri) = irX 2 fo
r„
(E
 E r ) + tT/2
(52)
+ 2kR
+ 4wR 2 (l  /„)
The radiative capture resonance is of the symmetrical BreitWigner
form (Eqs. 43 and 44). Because of the interference between capture
and potential scattering, the scattering resonance is not symmetrical
about E r . Instead, it goes through a minimum at an energy below E r
and falls off less rapidly on the highenergy side. The position of the
scattering minimum is given by
r, r„x D
*'** a* WW* (53)
and the cross section at the minimum is
<rmin(n,n) ^ 4xE 2 ( 1  /o + ^— j ) (54)
(assuming T n X/R <<C T and using Eqs. 37 and 52). Figures 6a and 6b
show typical shapes of scattering resonances for the case of target nuclei
of spin and spin •§• (indium). For the latter case, it must be kept in
mind that the capture scattering involves a definite, single J value of
the compound nucleus, while the potential scattering occurs for both
possible J values of the system (in this case J = §• ± ^ = 5 or 4), re
sulting in a finite scattering at the minimum. In addition, Fig. 6b cor
responds to a case where r « T y is> r„, and the presence of the radiative
capture process also prevents the scattering minimum from going to zero.
Another situation of interest is when r ~ T n i ^> T y , which is fre
quently encountered in nuclei of A < 50 for intermediateenergy neu
trons. In this case, application of Eq. (51) gives, for the position of the
scattering minimum,
r
E r  E min =  ctn fc (53a)
and, for the minimum cross section,
(Tmm(n,n) = ap (54a)
274 The Neutron [Pt. VII
In addition, the maximum of the resonance is shifted toward higher
energies:
r
#m a x  E r =  tan fc (55)
and the peak cross section is
<7 (n,n) = 4:irX 2 fi + <r p ' (55a)
For a scattering resonance involving a target nucleus of 7 = and S
scattering only (&> « £o) the scattering cross section will be zero at
Fig. 6a. Schematic drawing of the cross section through a slowneutron scattering
resonance for I = neutrons, for a target nucleus of spin 0. A spin / ^ of the
target nucleus would change this curve by multiplying it by /o(J) and adding to it
a constant cross section [1 — fo(J)]4:irR 2 .
the minimum (Fig. 6a). For 7 > or I > or both, the minimum will
not fall to zero, since the potential scattering for angular momenta not
associated with the resonance does not interfere with the capture
scattering.
Returning, now, to slowneutron reactions in heavy nuclei (Eqs. 52),
we note that the properties of a given resonance are completely deter
mined in terms of four parameters, E r , Y n , T y , and / . These can, in
turn, be ascertained from four measured quantities: (1) the resonance
energy, E r ; (2) the resonance width, T = T n + T 7 ; and (3), (4) the
peak capture (or total) and scattering cross sections, <To(n,y) and <r (n,n).
(We assume that R is given by r A Va X 10~ 13 cm.) Of these, c {n,n) is
usually most difficult to measure, since (for r 7 y> r„) it is only a small
fraction of the total. Thus, in a study of the 0.176ev Cd 113 (7 = ^)
resonance, Beeman (B12) has been able to extract all the parameters,
and to show that the level of the compound nucleus has the spin J = 1.
Sec. 2B]
Interaction with Nuclei
275
In the absence of a measurement of o ( n , n ), it is only possible to deter
mine the product / r„.
For pure scattering resonances (r « r„ 55> T y ) the statistical factor
/ is immediately determined from the difference between the peak cross
section and the minimum cross section, since c$ — tr m i n = 4irX 2 f.
20

I

yur J e = 4

J c =5 A
~
' ' ^^s.
^= 4 J
Peak Values of (T sc are
764 x 10" 24 cm 2 (,/„=• 4) and
626xl0" 24 cra 2 (,/ c = 5).
Resonance E r =1.44 ev.
Jl
J c =5
1 1
1 1 1 i
.90
2.70
E (ev)
Fig. 6b. The scattering cross section of indium (/ = %) for the 1.44 = ev reso
nance. The curves are computed from the known constants of the (capture) reso
nance and for the two possible values of J = 5 or 4. From Feshbach, Peaslee, and
Weisskopf (F27).
So far, we have discussed the case of a single resonance, neglecting the
effects of all the other resonances. This approximation is valid in the
vicinity of a single resonance when the level spacing D is large as com
pared to the level width r. If this is not the case, the levels can interfere
with each other, provided they belong to the same class (same I, J, and
parity). 1 Interference between levels has been discussed by a number
of authors (B25, B24, P10, B61, S37, W29).
Assuming, still, that there is only one resonance affecting the energy
region in which we are interested, the cross sections far from the reso
1 This statement applies only to the total cross section. Interference effects can
always occur in the angular distribution of the reaction products provided, only, that
the levels overlap.
276 The Neutron [Pt. VII
nance,  E — E r  » r, are
<r{n,a) = (56)
(E  E r f
a(n,n) ^ 4tR 2 (57)
These are of especial interest in the thermal neutron energy region when
the first resonance falls appreciably above or below thermal energies
(I E r I ^> E fh ). In this case
,rX 2 /r K r g
<rth(n,a) = — — — (58)
Recalling that X ~ E~ Yi and, from Eq. (37), that r„ ~ E*, we have,
provided that r„ is constant over the range of thermal energies,
(59)
where <r* is the cross section at a specific (arbitrary) energy E* (velocity
v*). This is the wellknown l/v law for slowneutron reaction cross sec
tions far from a resonance. Deviations from the l/v law will result from
too close proximity to a resonance; Eq. (56) shows that the effect of a
close positive energy resonance (E r > 0) is that the cross section falls
less rapidly than l/v, while a tooclose negative resonance results in a
thermal neutron reaction cross section which falls faster than l/v.
The thermal neutron scattering cross section will, on the other hand,
be constant if the first resonance is sufficiently far removed. However, in
the case of scattering, the effects of interference between the potential
scattering and the resonance will extend to considerable distances
(Figs. 6a and 6b), so that even a faraway resonance can have an appre
ciable effect on the thermal neutron scattering.
At the other extreme, we are frequently concerned with observations
of cross sections which represent an average over many energy levels;
this will be the case when the energy spread of the neutrons, used in the
measurement, is large compared to the level spacing of the compound
nucleus at the excitation energy involved. Assuming that X, r„, r are
essentially constant over the energy spread of the neutron source, the
average over Eq. (44) and sum over J = I ± % yield
2x 2 X 2 r„r„
a(n,a) = — (60)
If states of I > are possible, the above average must be summed over
all possible I values, each with its appropriate fi and appropriate neutron
width T n i and level spacing D;.
Sec. 2B] Interaction with Nuclei 277
If we consider heavy nuclei and 1 = interactions only, there are two
regions of interest: (1) r « r r » r„; here, as for thermal neutron cap
ture, <r(n,y) oc 1/v; (2) r ~ V n » Y y ; in this range, <r{n,y) oc 1/E. The
total cross section, however, has a more complicated energy dependence,
since it is now necessary to take into account the potential scattering
as well as the capture scattering and radiative capture (F28).
The more general approach developed by Weisskopf and coworkers
(F27, F28, W17, B34) gives, for the cross section for the formation of
the compound nucleus,
<r c « x(X + R) 2 P (61)
The factor 7r(X + R) 2 is a measure of the area of interaction of the neu
tron and the target nucleus; P is the probability that the neutron will
penetrate the nuclear surface. In the resonance region, P is a sharply
varying function of the neutron energy, being the resonance factor. In
terms of the simple picture which we have used, the incident neutron
wave, of wavelength X = 1/k, must join smoothly on to the neutron
wave, A = 1/K, at the surface of the nucleus. Since (for slow and
intermediate neutrons) K^> k, the amplitude of the neutron wave in
side the nucleus will, in general, be very small, except for certain ex
ceptional values of the neutron energy at which the amplitude of the
incoming wave at the nuclear surface has the maximum value. These
special cases (energies) correspond to the resonances, for which the
joining of the wave functions was schematically represented in Fig. 5.
We have previously (Eq. 21) given an expression for P which can be
interpreted as the average (over many resonances) probability for the
neutron to penetrate the target nucleus. Hence, the average reaction
cross section is
_ 4tt(X + R) 2 kK
c c ^ —  V (62)
(k + K) 2
For slow and intermediate neutron energies, X = 1/k » R, K = K >5> k,
whence
_ 4irX 2 & 4ir 500 , v
<T C = = ~ — rr barns (62a)
K kK E*
(for E in electron volts). Thus, the average cross section for the forma
tion of the compound nucleus also follows a 1/v law in these energy
ranges. Introducing the connection between neutron energy and neu
tron width from Eq. (37),
irT n K
k = (37a)
2D
278 The Neutron [Pt. VII
the average cross section for the formation of the compound nucleus
becomes
 2x 2 x 2 r n / ^
«*= ■ (63)
Since the cross section a(n,a) = <r c T a /T, Eqs. (63) and (60) are seen to
be equivalent, thereby confirming the interpretation of P (Eq. 21) as
the average over many resonances of the penetrability.
4. Summary ; Energy Dependence of Neutron Cross Sections. In the
preceding discussion we have outlined the basic considerations which
determine the interactions of neutrons with nuclei. We have observed
that the reactions of neutrons of a given energy with a given nucleus
depend on the level structure of the compound nucleus, at the excita
tion energy resulting from the neutron capture. Although we are, at
the present state of our knowledge, incapable of predicting the details
of the nuclear level structure, a great deal can, nevertheless, be said
concerning the average behavior of nuclei.
In referring to neutron energies, we shall continue to use the classifi
cation of Section 1B7 :
I. Slow neutrons: E < 1000 ev
II. Intermediate neutrons: 1 kev < E < 0.5 Mev
III. Fast neutrons: 0.5 Mev < E < 10 Mev
IV. Very fast neutrons: 10 Mev < E < 50 Mev
V. Ultrafast neutrons: 50 Mev < E
In addition to the dependence of nuclear level properties on the
kinetic energy of the incident neutron, the nuclear level spacing is very
different for nuclei of widely different atomic weight, A. It is therefore
convenient to classify nuclei according to the following system :
I. Light nuclei : A < 25
II. Medium nuclei: 25 < A < 80
III. Heavy nuclei: 80 < A
These dividing lines are, of course, not to be considered sharp bound
aries.
While we shall return, in a subsequent section, to the experimental
evidence on the question of the compound nucleus level spacing as a
function of the nuclear atomic weight, it is useful to bear in mind the
following summary of the average level spacing of the compound nucleus :
(1) Light Nuclei. For slow and intermediate neutrons, the levels are
very widely spaced, being ~10.1 Mev apart, on the average. Thus,
Sec. 2BJ Interaction with Nuclei 279
very few levels are encountered, and these are relatively broad and
easily resolved. In the fast and very fast neutron energy region, the
levels are closer, but still separate and resolvable.
(2) Medium Nuclei. For slow and intermediate neutrons, the levels
are ~1001 kev apart. Hence, few levels are encountered in the slow
neutron range, but many in the intermediate region. For fast neutrons
the level spacing rapidly decreases, and the levels begin to overlap. For
very fast neutrons the compound nucleus has a continuum of levels.
(3) Heavy Nuclei. In the slow and intermediate neutron ranges, the
level spacing is <~10005 ev. Many levels are observed for slow neu
trons; the levels are no longer resolved in the intermediate range; for
fast and very fast neutrons the compound nucleus has a continuum of
levels.
For ultrafast neutrons and for all nuclei, the compound nucleus pic
ture requires considerable modification.
(a) The Total Cross Section, c t . Most neutron cross section measure
ments involve the determination of the fraction of a given neutron beam
transmitted through a known thickness of material; such measurements
usually determine the total cross section, the sum of the cross sections
for all processes (absorption, scattering) which remove neutrons from
the beam,
at = 2 <r(n,a) (64)
a
For a neutron flux of <j> (neutrons • cm 2 • sec 1 ) incident on a slab of
material of density N (nuclei/cm 3 ) and thickness <(cm), the emerging,
undeviated flux <$> is
<j> = 4>o exp (Nta t ) (65)
Figure 7 shows typical slowneutron total cross sections for the three
categories of elements. The cross section of carbon exhibits thermal
neutron interference effects due to the crystal structure of the graphite
used in the measurement. Above ~1 ev, the cross section becomes and
remains constant. The cross section of cobalt shows a scattering reso
nance at 120 ev. The cross section in the thermal neutron region has a
1/v component due to radiative capture. The indium and iridium cross
sections show many resonances and large 1/v capture cross sections for
thermal neutrons. Owing to the close spacing of the levels, the neutron
width is small (Eq. 37) compared to the radiation width, so that radia
tive capture predominates in the observed resonances.
For intermediate and fast neutrons individual levels have been well
resolved only in light and medium nuclei; for these relatively large level
spacings, the neutron widths are much larger than the gammaray
280
The Neutron
[Pt. VII
e 3
1
c
<irai
shite

1
Sy
mbol on Reference No.
^igure in Legend
• 1
1
1
® 3
1 1 l 4 l 1
I 4
I A
2 4
2 4 7
I 4 7
2 4
0.01
E„(av)
100
(1) Columbia Velocity Selector (unpublished). (2) W. W. Havens, Jr., and L. J. Rainwater, Phys. Rev.,
75, 1296 (1949). (3) C. T. Hibdon and C. O. Muehlhause, Phys. Rev., 76, 100 (1949). (4) W. B. Jones,
Jr., Phys. Rev., 74, 364 (1948).
(a)
100
70
^ 40
en
 20
13 10
7
T
Co
O r*J
2,500 t
B.OOl) fc
arns
Symbol o
Figure
•
o
®
i Reference
in Legerj
1
2
No.
d
1
\
1
3
/
\
■I
\
\
/
""^.
>....
<A
•iu^.
y
J
\
/
'
..„
, r i
\
/
\
 — '
/
100
7 2
1000 4000
E n (ev)
(1) Wu, Rainwater, and Havens, Phys. Rev., 71, 174 (1947). (2) C. T. Hibdon and C. O. Muehlhause,
Phys. Rev., 76, 100 (1949). (3) F. G. P. Seidl, Phys. Rev., 75, 1508 (1949).
(b)
Fig. 7. Typical total cross section vs. energy curves for slow neutrons, from
Adair (A2). (a) Carbon, a light element. In this case, a t = <r sc  The variations,
below ~0.04 ev, are interference effects due to the polycrystalline structure of the
sample (see Section 5). (b) Cobalt, a medium element. Again, at ~ ct sc , except in
the thermal neutron range. Effects of scattering resonances at 120 ev and >2000
ev are evident.
Sec. 2B]
Interaction with Nuclei
281
20,000
10,000
In
2
1000
Symbol on Reference No.
Figure in Legend
1
2
3
® 4
o 5
1
4
<A
f
i
,
■1&T
V.
".
../
I
1 1
^100
b" 7
4
2
10
7
4
2
1
■".
FT' 
■
I'l J
£
]aj
I
v.
■ — i
u
_
4 7 2
0.1
2 4 7 2
1 10
B« (ev)
7 2
100
7
1000
(1) Borst, Ulrich, Osborne, and Hasbrouck, Phys. Rev., 70, 557 (1946). (2) Havens, Wu, Rainwater,
and Meaker, Phys. Rev., 71, 165 (1947). (3) B. D. McDaniel, Phys. Rev., 70, 832 (1946). (4) E. Fermi
and L. Marshall (unpublished). (5) C. T. Hibdon and C. O. Muehlhause, Phys. Rev., 76, 100 (1949).
(0)
4000
Ir
j\
1 1 1 1 1 1
2
11'
J 'l
Symbol on Reference No.
Figure in Legend
1000
tr
• l
x 2
7
...
/
1
" 2
S3
£ loo
—
?**
*.
l\
I
«**
f
f
b '
—
•
P
'■^
~~ S~
.
V
U
in
i
f
0.01
4 7 2
0.1
7 2
1 10
.E„(ev)
7 2
100
(1) Rainwater, Havens, Wu, and Dunning, Phys. Rev., 71, 65 (1947). (2) Sawyer, Wollan, Bernstein,
and Peterson, Phys. Rev., 72, 109 (1947). (3) Powers, Goldsmith, Beyer, and Dunning, Phys. Rev.,
53, 947 (1938).
(d)
Fig. 7 (Continued), (c) Indium, a heavy element. The famous resonance, at 1.44
ev, is well resolved. There are probably many unresolved resonances above 10 ev.
(d) Iridium, a heavy element. The first two capture resonances are fairly well re
solved. There are probably many unresolved resonances above 20 ev.
282
The Neutron
[Pt. VII
widths, and the observed resonances are due to scattering. We post
pone the discussion of scattering resonances and consider, instead, the
average behavior of the total cross section for intermediate and fast
neutrons. The average total cross section, according to the theory of
Feshbach and Weisskopf (Eqs. 49, 61, 62), depends on the properties of
the nucleus involved only through the nuclear radius, R. 1
ft!
2.0
1.0
1 1
" Uo = B
.1
4
 y
_\

As
V
s 2 *"
\
cje+i
) 2

<^
ii;
^ _
—

123456789 10
x — kR
Fig. 8. Total cross sections (averaged over resonances) as a function of x = kR,
for Xo = KqR = 5 and 8. <ro is the total cross section for an impenetrable sphere
of radius R. The broken curve gives the approximate behaviors for large x. From
Feshbach and Weisskopf (F28).
Figure 8 is a plot of the expected average total cross section vs. neutron
energy. The cross section is given in units of wR 2 , and the energy in
units of
x = kR = 0.222RE' A (66)
where R is in units of 10~ 13 cm and E in Mev. The curves are plotted
for two values of X = K R: X = 5 (A « 40) and X = 8 .(A ~ 160).
The curve labeled <r is the total cross section of an infinitely repulsive
1 The parameter K «1X 10 ~ 13 cm, the wave number of a neutron in the nucleus,
is assumed to be the same for all nuclei; this is not strictly true, but the variations
in Ka are expected to be small.
Sec. 2B] Interaction with Nuclei 283
sphere of radius R. For large values of x, the curves are approximately
fitted by the broken curve,
«r« « 2t(R + X) 2 (67)
It is important to note that, while the value of the total cross section
approaches 2icR 2 for X « R, the deviations from this asymptotic value
are quite appreciable, even for very fast neutrons; thus, for E = 50
Mev, X = 0.64 X 10 13 cm, which cannot be neglected.
In Fig. 9 the experimental values of the total cross section are com
pared with the theory for a number of elements: (a) iron, A « 56;
(b) silver, A ~ 108; (c) antimony, A « 122; and (d) lead, A « 207.
The measurements employed relatively poor resolution, and as a result
the averaging process was, to a large extent, performed experimentally.
The agreement with theory is seen to be excellent, except in the case of
antimony, where the observed cross section falls considerably below the
theoretical value at low energies. 1
(b) Cross Section for Formation of a Compound Nucleus, a c . The cross
section for the formation of a compound nucleus includes all reactions,
except elastic scattering in which the quantum state of the nucleus is
unchanged. It is usually defined as
<?c = O t — (T e l (68)
While this definition does not accurately take into account the effects
of interference between capture and potential scattering, such effects
are essentially eliminated when the cross section is averaged over many
resonances.
For light and medium nuclei, a c for slow and intermediate neutrons
is mainly due to capture scattering (except for thermal and slower neu
trons). For heavy nuclei, in these energy regions, a c is almost entirely
due to radiative capture. For fast and very fast neutrons, elastic scat
tering contributes little to the value of <r c , the other possible reactions
— inelastic scattering, (n,p), (n,a), (n,2n), etc. — predominating.
The energy dependence of a c (averaged over resonances) is described
by Eqs. (61), (62), (62a) and plotted in Fig. 10. (The method of plot
1 This "anomalous" type of behavior has been found, by the Wisconsin group
(M31), to occur with unanticipated frequency. Furthermore, extension of the total
cross section measurements to 3.2 Mev has brought to light, for the heavier ele
ments, the frequent existence of a broad maximum in a t which appears to move to
higher energies with increasing mass number. These phenomena are strongly sug
gestive of the scattering by a finite potential, and could possibly be interpreted as
supporting an "independent particle" model for fast neutron scattering. It is, at
present, difficult to see how these results can be reconciled with the apparent wide
spread applicability of statistical models.
284
The Neutron
[Pt. VII
Fe
10
n
_
OS
N 6
"
k
/i
4
"I
mi
"i
^
"~fV
Theory
V
),
?,
Experin
ent
i
i
t
i
1
1
1
1
1
02 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.1
x=kR
(a)
t3
Ag

~"T
 _ ^
eory

Exp
J
iriment
P^
rV '° sr \7
i
1
I
i
1
1
1
1
1
i
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
x=kR
(b)
Q5
Sb
~^L
0.2 0.4 0.6
0.8 1.0
x=kR
1.2 1.4 1.6 1.8 2.0
(C)
Fig. 9. Comparison of the theory of Feshbach and Weisskopf (F28) with measured
total cross sections for a number of elements. The measurements are due to Bar
schall and coworkers (references in F28). (a) Iron, A =< 56. (b) Silver, A « 108.
(c) Antimony, A «= 122. In the case of antimony the observed cross section is in
better agreement with the prediction of scattering from a rigid sphere.
Sec. 2B]
Interaction with Nuclei
285
8
Pb
6
~~~
— ~v_
. Theory
<*

Experiments^
2
i
i
'

1
1
1
1
1
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
x=kR
(d)
Fig. 9 (Continued), (d) Lead, A = 207.
Sd.b
2.4
2.2
2.0
1.8
1.6
1.4
1.2
I
Vo =
5
\
•
*) 2
V
^0=8
^ "^
 :;:: ^=c:
^^5.
 —
.
1.0
^cii
10
Fig. 10. Cross section for the formation of a compound nucleus vs. x = kB, accord
ing to Feshbach and Weisskopf (F28). The solid curves are for X = KoR = 5, 8,
and 11. The broken curve, appropriate to large x, is <r c « ^a t « ir(iJ + X) 2 .
286 The Neutron [Pt. VII
ting is the same as that employed in Figs. 8 and 9.) As seen from Eq.
(62), a c > ir(R + X) 2 for x » X ; the deviation of <r c from w(R + X) 2
depends on the value of x/X , being «11 percent for x = X and «4
percent for x = 2X . For large values of x, <y c ~ \<j t and, since cap
ture elastic scattering is negligible for large x, a c ~ a e i; i.e., for large x,
the total cross section divides into approximately equal parts elastic
scattering and capture.
C. Types of Neutron Reactions
In the previous section, we have discussed the general properties of
neutron reactions and the formation of the compound nucleus. Although
it has been observed that the nature of the compound nucleus deexcita
tion is governed by the competition between the various possible modes
of decay, the properties of this competition have been treated in the
most general way. This section is devoted to the anatomy of neutron
reactions — to the details of the competition between the various possible
neutron reactions, as they are influenced by the properties of the nuclei
involved and by the energy of the impinging neutrons.
The relative probability of a given neutron reaction (n,a) — neutron
in, particle a out — is defined in terms of the cross section <r(n,a). For a
(pure) material containing NV nuclei in a neutron flux <£, the number of
(n,a) reactions per second is NV<f>a(n,a) .
1. Scattering (n,ri). Neutron scattering is one of the two reactions
which are energetically possible for all nuclei at all neutron energies.
(The other is radiative capture.) The observed elastic scattering is, in
general, the result of a superposition of the potential scattering and
the capture scattering (Eq. 51). However, this superposition is such
that it is coherent (addition of amplitudes, interference effects) for
the capture scattering and that part of the potential scattering which
involves the same neutron orbital angular momentum, Zh, and the same
total angular momentum state, /. The remainder of the potential scat
tering is observed as an incoherent background (addition of cross sec
tions) to the scattering, as described by Eq. (51).
(a) Slow Neutrons. In the slowneutron energy range, the neutron
wavelength is very much larger than any nuclear radius. Hence, neu
trons can react (with appreciable probability) only in the I = state.
Thus, the potential scattering cross section is 4irR 2 (Eq. 49a), and the
scattering cross section reduces to Eq. (52).
In the absence of nuclear resonances, the slowneutron (potential)
scattering cross section would be expected to increase with atomic num
ber as A % . Table 5 is a compilation of observed slowneutron scatter
ing cross sections together with the corresponding values of 47r.fi! 2 , com
Sec. 2C]
Interaction with Nuclei
287
puted on the assumption R = 1A7A H X 10 13 cm. The scattering
cross sections, in many cases, deviate considerably from 4irR 2 , being
both larger and smaller. These deviations are too large to be ascribed to
fluctuations in R. An additional, noteworthy fact is that the deviations
occur, with comparable frequency, at all values of the atomic number.
These fluctuations in <r sc can be ascribed to the effect of a nearby
resonance, being the result of the interference between the potential and
the resonance scattering. A value of <r sc > 4xi2 2 indicates that the
closest resonance is at an energy lower than that at which the cross sec
tion was measured (frequently E r < 0), while <r sc < 4irR 2 indicates a
resonance at a higher energy.
The probability that a scattering measurement, made at a given
(arbitrary) neutron energy, will fall on a resonance is inversely propor
tional to the level spacing. That is, the average distance of a given
energy from a resonance is proportional to the level spacing; thus, the
larger the level spacing, the farther, on the average, from the closest
level. However, the region over which interference effects are impor
tant is also proportional to the level spacing (Eq. 53). Hence, the
probability that a scattering measurement will give a cross section dif
ferent from AwR 2 is essentially independent of the level spacing and,
therefore, of the atomic number of the scatterer.
TABLE 5
Slow Neutron Scattering Cross Sections for Free Nuclei
Element
At
Csc t
(barns)
4irfl 2 §
(barns)
E  (ev)
References If
iH 1
20.36
W10, M24
iH 2
3.3
th
S33
2 He 4
1.4
(0.2)
th
H41
3 Li 7
~1.5
0.99
th
S33
4 Be 9
6.1
1.17
(0.8)
th
S33
B B
11, 10
3.9
1.33
(1.5)
C
W10
6 C 12
4.70
1.42
(1.8)
th
W10
7 N U
9.96
1.58
10200
W10, M25
8 16
3.73
1.70
(2.3)
151000
W10, M25
9 F 19
3.3
1.93
0.2540
W10, R5
288
The Neutron
[Pt. VII
TABLE 5 (Continued)
Slow Neuteon Scattering Ceoss Sections toe Free Nuclei
Element
At
<Tsc t
(barns)
4irj? 2 §
(barns)
E  (ev)
References If
ioNe
20
2.4
2.01
th
H41
u Na 23
3.3
2.20
1800
H58
i 2 Mg
24, 26, 25
3.9
2.28
(2.5)
th
S33
«A1»
1.35
2.44
(2.7)
th
W10
uSi
28
2.2
2.51
1100
W10, R5
i 6 P 31
~3.3
2.68
110
G20
16S
32
~1.2
2.74
(2.1)
10400
A2
nCl
35,37
14.2
2.93
(2.8)
th
H59
isA
40
0.8
3.17
th
H41
i 9 K
39
~2
3.13
th
S33
2oCa
40
3.3
3.18
th
S33
2lSc«
12.8
3.44
th
H40
2 2 Ti
48
6
3.58
th
S33, F27, G12
23V 61
5.02
3.73
th
H26
24Cr
52
3.7
3.78
th
S33
24Cr 63
8.4
3.83
th
H40
26 Mn 56
2.1
3.93
th
S33
26 Fe 54
2.4
3.88
th
S33
2 6 Fe 56
12.5
3.97
th
S33
26 Fe 67
2
4.02
th
S33
2eFe
56
11.3
3.97
th
S33
11.1
(3.9)
1.44
F27, H36
27C0 69
5
4.12
C
F27, B57
28 Ni 68
27
4.07
th
S33
28Ni 6 °
1.0
4.16
th
S33
28 Ni 62
9
4.25
th
S33
28 Ni
58, 60
16.7
4.10
th
S33
29CU
63, 65
7.6
4.33
th
S33
8.3
(3.8)
1.44
F27, H36
3oZn
64, 66, 68
4.1
4.41
(4.4)
th, 1.44
S33, F27, H36
aoZn 67
7
4.48
th
H40
3iGa
69, 71
3
4.60
th
W10
32Ge
74, 72, 70
8.3
4.71
th
S33
33AS 76
~7
4.83
th
S33
Sec. 2C]
Interaction with Nuclei
289
TABLE 5 (Continued)
Slow Neutron Scattering Cross Sections for Free Nuclei
Element
A t
08C X
(barns)
4ttK 2 §
(barns)
E  (ev)
References f
34 Se
80, 78
10
5.00
(5.0)
C
F27, G12
ssBr
79, 81
5.9
5.04
th
S33
3eKr
84, 86, 82, 83
7.2
5.20
th
H41
37 Rb
85,87
5.4
5.27
th
S33
3sSr
88
9.3
5.36
th
S33
39Y 89
3.9
5.41
th
H40
40 Zr
90, 94, 92, 91
~7
5.50
th
S33
4lNb 93
6.1
5.58
th
S33
42 Mo
98, 96, 95, 92
7.3
5.69
th
S33
44R11
102, 104, 101,
99, 100
6
5.92
C
F27, G12
4 5 Rh 103
3.5
5.96
th
H40
46 Pd
106, 108, 105,
4.7
6.11
th
S33
110
4.4
C
F27, G12
4 7 Ag 107
10
6.12
th
S33
47Ag 109
6
6.20
th
S33
4?Ag
107, 109
7
6.15
(5.8)
th
S33
4 8 Cd
114, 112, 111,
110, 113
5.3
6.32
(6.5)
5100
G20
49ln
115
2.2
6.42
th
W10
soSn
120, 118, 116
4.8
6.56
(6.9)
th
S33
6 iSb
121, 123
4.1
6.67
(6.7)
th, 0.12
S33, G20
5 2 Te
130, 128, 126
5
6.88
C
G12
53I 127
3.7
6.86
th
S33
5 4 Xe
132, 129, 131,
134
4.3
7.02
th
H41
65 Ce 133
~7
7.08
th
S33
56Ba
138, 137
8
7.23
C
G12
69 Pr 141
7.9
7.35
th
H40
62Sm
152, 154, 147,
149, 148
23
7.68
th
H40
63EU
153, 151
29.7
7.73
th
H40
64Gd
158, 160, 156,
157, 155
26
7.90
th
H40
290
The Neutron
[Pt. VII
TABLE 5 (Continued)
Slow Neutron Scatteeing Cross Sections for Free Nuclei
Element
A\
f«c t
(barns)
4xfl 2 §
(barns)
E  (ev)
References H
72H1
180, 178, 177,
179
25.7
8.61
th
H40
73 Ta 181
6.9
8.69
th
S33
74W
184, 186, 182,
183
5.6
8.78
th
S33
74 W 186
23
8.84
th
H40
76 0s
192, 190, 189,
188
~10
8.98
C
G20, G12
7sPt
195, 194, 196
11.1
9.14
th
S33
79 AU 197
~9
9.19
(7.1)
th
S33
soHg
202, 200, 199,
15
9.30
0.110
G20
201, 198
21.5
(8.7)
th
H60
81T1
205, 203
9.6
9.42
0.11
F27, G20
82 Pb
208, 206, 207
11.5
9.51
th
S33
12.4
(7.6)
110
F27, G20
83 Bi 209
9.2
9.56
510
F27, G20
10
(7.8)
th
S33
92U
238
8.2
10.5
th
U2
f When more than one isotope is involved, this column lists the most impor
tant isotopes, of relative abundance greater than 10 percent, in the order of
abundance.
t The cross sections are for free nuclei. Most observations have been made on
bound nuclei; these have been corrected according to the relationship
°"free
C*i)'
^bound
§ Measured nuclear radii, taken from the collection of Blatt and Weisskopf
(B34), are given in parentheses. Otherwise they are computed on the basis of
R = 1.474H X 10 13 cm
 The symbols have the following meanings: th, a thermal Maxwell distribu
tion of neutrons at a temperature of <~300°K; C, those neutrons, in a thermal
distribution, capable of penetrating through appreciable thicknesses of cadmium.
(These have energies above the cadmium cutoff, ~0.5 ev.)
f This compilation leans heavily on the previous compilations of Shull and
Wollan (S33), Way at al. (W10), and Feshbach, Peaslee, and Weisskopf (F27),
in which references are given to the original investigations.
Sec. 2C]
Interaction with Nuclei
291
However, the curves of c sc vs. E will be very different for nuclei of
widely different atomic number, since the number of resonance and
their characteristics depend strongly on the level spacing.
(1) Light Nuclei. In general, the main reaction of slow neutrons
with light nuclei is elastic scattering. There are four exceptions — Li 6
and B 10 , for which the main reactions are (n,a); He 3 and N 14 , for
which there is an appreciable slowneutron (n,p) cross section. Aside
from these, a sc = v t (see Fig. 7a).
(2) Medium Nuclei. With regard to the scattering of slow neutrons,
medium nuclei are similar to light nuclei, except that resonances are
sometimes encountered. These are primarily scattering resonances
since the level spacing is quite large. For this case r ~ r n > r T , the
peak cross section
2.6/
<r (n,n) ^ 47rX r 2 / = X 10 6 barns (55a')
E r
(E r is in electron volts.) The 120ev resonance in cobalt (Fig. 7b) is an
example of such a scattering resonance. The peak cross section for this
resonance should be ~13,000 barns since, for Co 59 , I = f(/~ f).
The smallness of the peak cross section in Fig. 7b, as well as the absence
of the expected interference between the resonance and potential scat
tering, is due to the poor resolution of the measurement. Since the
TABLE 6
Properties of Some Neutron Scattering Resonances
Com
Target
Nucleus
Spin
/
E r (ev)
r„ (ev)
r„/r
pound
Nucleus
Spin J
Observed
<r (barns)
4xX r 2 /
(barns)
References
nNa 23
3
2
3,300
340
~0.999
2
550
540
H58, t
16 S 32
111,000
18,000
~l
1
2
21.8
23.4
Al
17C1 36
2
75
2.63 fl
0.90 t
(2)
H59
2 6 Mn 55
345
20
0.990
3
45,000
4,400
S17, H40, H43
27C0 59
7
2
123
»4
0.94
(4)
12,500
12,200
H38, S17, H40, t
33AS 76
3
46
0.11
0.72
~28,000
t
69ft 141
5
1
(10?)
0.93
(~130,000)
H40
6 2 Sm 152
8.2
~0.3
0.66
1
2
320,000
H40
74W 186
19.25
0.25
0.62
1
2
~90,000
135,000
H39, H40, S19
8lT1 eon)
1
260
«3.2
0.52
(1)
7,500
H40
83Bi 2M
"Z
770
3.5
~1
~1,700
t
t Some of the original data in this table have been revised in the light of recent measurements by the Harwell time
offlight velocity selector group (M26) and by the Argonne fast chopper group (whose members included L. M. Bol
linger, R. R. Palmer, and S. P. Harris). We are grateful to these groups for private communications of their results.
t These values are at E = \ E r \ =75 ev.
292
The Neutron
[Pt. VII
recognition, by Goldhaber and Yalow (G14), that the resonance of
manganese at ~300 ev is a scattering resonance, many such resonances
have been observed and investigated in medium (and some heavy)
nuclei. Table 6 summarizes the properties of a number of scattering
resonances.
(3) Heavy Nuclei. In heavy nuclei, with their relatively small level
spacings, the gammaray width is generally larger than the neutron
Symbol o
Figure
1 1
l Reference No.
in Leoend
1
® 1
oA 2
:

4
1
k~
L
«h <
o
i
^
®
/ 1
//
•V
i
®
V
9 .
l*C
/
***•&
w
A
A
A
'
0.2 0.4 0.G 0.8 1.0 1.2 1.4 1.6 1.8
£„(Mev)
(1) Fields, Russell, Sachs, and Wattenberg, Phys. Rev., 71, 508 (1947). (2) Adair, Barschall, Bockel
man, and Sala, Phys. Rev., 75, 1124 (1949). (3) Bockelman, Miller, Adair, and Barschall, Phys. Rev.,
84, 69 (1951). (4) Freier, Fulk, Lampi, and Williams, Phys. Rev., 78, 508 (1950).
Fig. 11. Total (scattering) cross section of O 16 for intermediate and fast neutrons,
showing wellresolved Pwave resonances at 0.440, 1, and 1.3 Mev; from Adair
(A2). Measurements from ~1.8 to 4 Mev have exhibited additional resonances
whose angular momenta and parities have been deduced by Baldinger, Huber, and
Proctor (B5) by means of accurate measurements of the angular distributions of the
elastically scattered neutrons.
width, r ~ T 7 > r„. Accordingly, the peak scattering cross section is
small compared to the peak radiative (and total) cross section,
<ro(n,n) = ~ <r (n,y)
(47a)
The resonance scattering of cadmium (B12) is an example of this case.
Owing to fluctuations in the nuclear level spacing, and to the differ
ences in neutron binding energies, levels are sometimes encountered in
heavy nuclei for which T n > F y . The broad resonance of W 186 at 19 ev
has been found to be such a scattering resonance (H39). 1
1 The existence of broad, scattering resonances for slow neutrons on heavy nuclei
might be additional evidence for the necessity, under certain circumstances, of re
garding neutron reactions from the independent particle point of view.
Sec. 2C]
Interaction with Nuclei
293
(sureq) % D
u
o
<y
&H
d
E
X
Ei
CD
d
'55
(H
T3
fe
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(=1
o
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d
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ffl
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^
t.
T3
to
=3
>
.id
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Eh
d
FH
ol
0)
CO
d
rd
N
,d
BQ
(1
bl)
c«
d
O
d
o
£
a
o
ro
o.
o
d
o
Eh
03
r/>
0>
fl
0)
11
CQ
+s
h
d
>
d
3
o
n
d
a>
0J
a;
1— I
o
J4
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03
t3
*Q
H
Eh
3
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0J
+3
E
o
to
<
ri
u
<
u
03
3
0)
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r^
M
o
O
a
CO
V)
.2
o
£
op
O
o
o
go
d
03
en
CO
I7J
v
;__;
a
O
03
03
a
M)
V)
o
d
0>
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d
t>
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tf
+■*
si
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d
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H
CJ
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hr
m
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.fi
294
The Neutron
[Pt. VII
(siueq) ?x>
Sec. 2C]
Interaction with Nuclei
295
(b) Intermediate Neutrons. The intermediate neutron energy region
is, with respect to the interaction of neutrons with nuclei, a transition
region. At the lowenergy end, the neutron wavelength is still large
V
i1
'
V
\
» v
VVi
\\
A
1
l\
y v
\> s
f\A' y )n
■\h[
Ws
o°A
o
o
A
A
0.4 0.6
£7„(Mev)
1.0
Fig. 13b. Scattering cross section for intermediateenergy neutrons of V 51 ; data
from Blair and Wallace, Phys. Rev. 79, 28 (1950).
S 3
s
n
n
Symbol o
Figure
n Reference N
in Legend
1
•1
1
If
J
L
•
3
/
/
/
i
'1
\
\i
'■Ahh
{
Ute
i
/
v\
«
\i
\
i
A
I
A
0.7 0.E
1.2 1.4
E„(Mev)
1.6
1.8
(1) Peterson, Barschall, and Bockelman, Phys. Rev., 79, 593 (1950). (2) Freier, Fulk, Lampi, and
Williams, Phys. Rev., 78, 508 (1950). (3) Fields, Russell, Sachs, and Wattenberg, Phys. Rev., 71,
508 (1947).
Fig. 13c. Scattering cross section of S 32 in the fastneutron region; from Adair (A2).
compared to nuclear radii ; at the upper end, the neutron wavelength is
comparable with (and, for heavy nuclei, smaller than) nuclear dimen
sions. In this region, the character of the neutron scattering process
slowly changes.
(1) Light Nuclei. Figure 11 shows the scattering cross section of
oxygen (7 = 0) in the intermediate (and lower end of the fast) energy
range. The cross section is constant up to the resonance at 440 kev.
296 The Neutron [Pt. VII
This resonance must be ascribed to capture of I = 1 neutrons (P reso
nance), since its peak cross section (14 barns) is greater than 47rX r 2 = 6
barns, and since it shows none of the interference properties associated
with <S resonances. The value of the peak cross section is consistent
with a value J = J (/i = 2) for the compound nucleus; i.e., a a (n,n)
= 87rX r 2 + <r p ' = 12 + 4 = 16 barns. The resolution of the measure
ment is excellent (~10 kev), but its finite value is sufficient to account
for the difference of ~2 barns between the observed and expected peak
cross sections.
(2) Medium Nuclei. In Fig. 12 is plotted the total (scattering) cross
section for sulfur {A = 32). This curve is especially interesting, since
it illustrates many of the features of scattering resonances which we
have previously discussed. The broken curve is a plot of 4xX 2 . The
three *Swave resonances, at 111 kev, 375 kev, and 700 kev, have peak
cross sections close to 4ttX 2 , indicating the excellent resolution of the
measurements. They show very well the interference between the po
tential and resonance scattering. The other six resonances appear to be
due to P(l — 1) or D(l = 2) neutrons, and are incompletely resolved.
Of these, the 585kev resonance can be ascribed to a / = f state of the
compound nucleus, S 33 (P13).
(3) Heavy Nuclei. For heavy nuclei in the intermediateenergy re
gion, the resonances are usually much more closely spaced than the
resolution of the best available measuring devices. At the lowenergy
end the resonances are still primarily due to radiative capture (T y > T n ),
while at the upper end the neutron width has caught up with the gamma
ray width, and scattering predominates. Toward the highenergy end
of the intermediate region inelastic scattering starts to compete as a
possible reaction, but we postpone the discussion of this reaction.
For certain of the heavy nuclei, the level spacing of the compound
nucleus, at the excitation energy corresponding to neutron capture, is
anomalously large. Such appears to be the case for the magic number
or near magic elements lead and bismuth, owing either to an anomalously
small neutron binding energy, or to an intrinsically large level spacing, or
both. The cross section of Pb 206 is shown in Fig. 13a. As an interesting
comparison, we also plot the cross section of V 51 in the intermediate
region (Fig. 13b), and that of S 32 in the fastneutron region (Fig. 13c).
All of these show approximately the same level spacings.
The curves of Fig. 13 can be used to illustrate some other interesting
possibilities in scattering resonances. If the resonances are still resolva
ble at relatively large values of x = kR >" 1 (due either, as in S 32 , to
large spacing for fast neutrons or, as in Pb 206 , to a large nuclear radius),
I = 1 potential scattering becomes appreciable, and Pwave resonances
Sec. 2C] Interaction with Nuclei 297
are no longer expected to be symmetrical. Thus, the 1055kev sulfur
resonance must, because of the large peak cross section, be ascribed to
neutrons of I > 0; however, it also shows a definite interference mini
mum, due to the Pwave potential scattering. It is therefore a Py 2 reso
nance.
Another interesting anomaly occurs in the region £ = % ~ v/2.
(&>o ~ f/2, for which the same considerations apply, requires neutrons
of much greater energy.) For an <Swave resonance in this energy region,
application of Eqs. (53a) and (55) indicates that the minimum cross
section, o min = a p ' , occurs at E min = E r , while the maximum disappears
(i.e., E/max — »• °°). This leads to crosssection curves which have the
appearance of inverted resonances, as observed in the Pb 206 cross sec
tion of Fig. 13a.
(c) Fast Neutrons. Of the total cross section for fast neutrons, ap
proximately half involves the formation of a compound nucleus; the
rest is due to elastic (potential) scattering, in which the neutron passes
close to, but not into, the target nucleus.
I. Capture Scattering and Inelastic Scattering
With the exception of a few light nuclei (and the heaviest nuclei) the
most probable result of fastneutron capture is neutron reemission. As
long as the incident neutron energy is less than the energy of the first
excited state of the target nucleus, the neutrons will be reemitted with
their incident energy (minus, of course, the energy lost to the recoil
nucleus to conserve momentum). As soon as the incident neutron
energy exceeds the energy required for the excitation of the lowest level
of the target nucleus, it becomes possible for the product nucleus to be
left in the excited state; correspondingly, the emitted neutron will have
a smaller kinetic energy than the incident neutron. This process is
known as inelastic scattering.
From the point of view of the excited compound nucleus, decay to the
ground state (capture elastic scattering) and to the various energetically
possible excited states (inelastic scattering) of the product nucleus are
competing processes. Each mode of decay is characterized by a partial
width, r,, which is a measure of its relative probability. However, from
the experimental viewpoint, there are, in addition to the difference in
energy of the elastically and inelastically scattered neutrons, two dis
tinguishing features between the two processes: (1) Capture elastic
scattering and potential scattering are coherent processes, and the elas
tic scattering cross section can therefore exhibit interference effects of
the type previously discussed. (2) Inelastic scattering leaves the prod
298 The Neutron [Pt. VII
uct nucleus in an excited state, from which it decays by the emission of
one or more gammarays. Indeed, it was through the observation of
this gammaradiation that the inelastic scattering process was dis
covered (Lll, K12). There have been a number of subsequent investi
gations of inelastic scattering by the observation of the resulting gamma
radiation (A29, S12, B13, G27); however, all but the most recent of
these have been too crude to permit detailed interpretation.
The most useful information concerning inelastic scattering has been
derived from experiments involving the observation of the energy dis
tribution of inelastically scattered neutrons. The interpretation of such
experiments has been discussed by Feld (F9).
In the ideal inelastic scattering measurement, a monoenergetic neutron
beam would be scattered from a thin target of the element under in
vestigation; the scattered neutrons would have a line spectrum, each line
corresponding to the excitation of a given level of the target nucleus.
The energies of the inelastically scattered neutron groups would give
the positions of these energy levels, and the relative strengths of the
groups would be a measure of the relative values of the partial widths
for the corresponding compound nucleus decay. In addition, the angu
lar distribution of a given group of inelastically scattered neutrons (with
respect to the direction of the incident neutrons) would yield valuable
information concerning the angular momentum and parity properties
of the levels involved. Finally, each group of inelastically scattered
neutrons is accompanied (in coincidence) by one or more gammarays;
the correlation between the direction of emission of the neutrons and
gammarays would yield further information concerning the angular
momentum and parity properties of the levels.
An experiment of the type outlined above is, in practice, exceedingly
difficult, since it requires, in addition to a monoenergetic neutron source,
a fastneutron detector of high resolution. Most attempts to measure
the energy distribution of inelastically scattered neutrons have em
ployed "threshold" detectors, i.e., a detector sensitive only to neutrons
of energy above a fixed value, E t . Since, in most cases, the sensitivity
function of the detector is not well known, such experiments are usually
difficult to interpret unambiguously.
Many inelastic scattering measurements have employed heteroener
getic neutron sources, such as the RaaBe or RaaB source, and thresh
old detectors. Owing to the extreme difficulty in interpreting such ex
periments (S65), they will be omitted from this discussion.
(1) Light Nuclei. The lowlying levels of light nuclei are very widely
spaced (~0.55 Mev). Hence, fast neutrons can excite few levels. In
many cases, for neutrons of a few Mev, only one level of the target
Sec. 2C] Interaction with Nuclei 299
nucleus will be available for excitation. The inelastic scattering cross
section will depend in detail on the properties of this level, according to
the following general considerations: Let <r c be the cross section for the
formation of a compound nucleus. 1 The cross section for the excitation
of the level (through inelastic scattering) is
<r z = — (42a)
Ti, the partial width for inelastic neutron emission, is given by Eqs.
(37) to (40) :
T x « (21 + ^CDiE^vi^R) (38a)
in which the symbols have their usual meanings; E\ and ki are the
energy and wave number of the (inelastically) scattered neutron :
E^E  D x (33a)
(E is the incident neutron energy; D x is the excitation energy of the
target nucleus level involved.) The total width
r = r + t 1 + r a + r 6 + • • • (28a)
includes the width for elastic neutron reemission, r , and the widths
for decay by emission of any other particles, r , Tb, etc.
In the expression for a\ (Eq. 42a) all the factors are, to a greater or
lesser extent, energy dependent. However, in the immediate neighbor
hood of the threshold (E « Dx), the main energy dependence arises
through the factor Ei A vi{kiR) in IV (It is clear that, for energetic
reasons, Ti = for E < D x .) If we consider a region close to the
threshold, such that E x <<C E, the factors a c and r are essentially con
stant. For inelastic scattering in which Swave (I = 0) neutrons are
emitted, the cross section will have an E^ A energy dependence near the
threshold; for emission of I > neutrons, the cross section near thresh
old will increase as E^ l+y2) .
The angular momentum of the emitted neutrons is determined by the
values of the spins and parities of the initial (compound nucleus) and
final (product nucleus) states. In many cases these values are such that
neutrons of only a single, definite angular momentum can be emitted.
In some cases, however, the spin values are such that angular momen
tum conservation could be satisfied in a number of alternative modes of
decay, i.e., more than one I value is possible for the emitted neutron.
1 The energy dependence of a c has been discussed on page 283. It should, however,
be borne in mind that the general considerations, used to derive Fig. 10, are of ques
tionable validity for light nuclei in the fastneutron range.
300 The Neutron [Pt. VII
In such cases the parity selection rule further limits the number of pos
sible I values; e.g., if the parity of the initial state is the same as that of
the final state, neutrons can be emitted only with even I values; for
states of opposite parity, only odd I values are permitted. Consequently,
and also because of the strong energy dependence of Ti, one — the lowest
— of the possible I values of emitted neutrons will be predominant in
most cases.
When the angular momentum of the emitted neutrons has a definite
value and if, furthermore, the absorption of the incident neutrons to
form the compound nucleus state involves a single I value, the inelasti
cally scattered neutrons will have a definite and predictable angular
distribution with respect to the direction of the incident neutrons. The
angular distributions, in a number of possible cases, have been given by
Hauser and Feshbach (H44) .
As an example of the type of angular distribution that could be en
countered, we give the result for a specific (hypothetical) case:
Target nucleus: I = \
Compound nucleus : J = 3
Product nucleus: /'=■§•
Incident neutron: I = 2
Scattered neutron: V = 1
The resulting angular distribution of (inelastically) scattered neutrons is
W(e) = i + f cos 2 e
(0 is the angle between the incident and scattered neutrons.)
Similar information concerning the three states involved could be
obtained by observation of the angular correlation between the inelasti
cally scattered neutrons and the gammaradiation which follows the
inelastic scattering. In the example given above, the transition AI = 1
with parity change would be achieved through the emission of an electric
dipole (or magnetic quadrupole) gammaray. Angular correlations in
successive particle gammaray emission are discussed in Part IX of
Volume III.
As previously indicated, many cases of inelastic scattering will be
considerably more complicated than the example discussed above. For
instance, it may be possible for the incident neutrons to excite more
than one compound nucleus state, requiring a number of incident
neutron I values. Correspondingly, the decay of the compound nucleus
to the excited product nucleus can also involve a number of different
Sec. 2C] Interaction with Nuclei 301
angular momenta, and the angular distributions for the different I
values can interfere. If this situation prevails, the calculation of the
angular distribution of the inelastically scattered neutrons requires
further knowledge of the details of the competition. Some of these
possibilities have been considered by Hauser and Feshbach (H44) and
by Wolfenstein (W38) :
There has been very little experimental investigation of the inelastic
scattering of fast neutrons by light nuclei. The work of Beghian, Grace,
Preston, and Halban (B13, G27), on the inelastic scattering of 2.5Mev
neutrons by beryllium, carbon, fluorine, magnesium, sulfur, chromium,
iron, and copper is of interest in connection with the above considera
tions. They detected the inelastic scattering by observing the resulting
gammaradiation and used the intensity of the gammarays as a meas
ure of the cross section. In the cases of carbon and beryllium they
were unable to detect any gammaradiation. For the other elements
they observed a single (monoenergetic) gammaray in the first four
cases, indicating that only a single level of the target nucleus was in
volved; the last two yielded complex gammaray spectra. These results
are summarized in Table 7a.
The observed cross sections, when combined with a knowledge of <r c ,
yield values of the relative probability for inelastic scattering Ti/T
(Eq. 42a). These are shown for fluorine, magnesium, and sulfur in
Table 7b, together with the data from which they were derived. The
values of cr c have been estimated from the available data (A2, W32).
In the last column of the table, we also give theoretical values of Ti/T,
based on the assumption that only elastic and inelastic scattering are
of importance (r = r + 1^), and that both types of scattering in
volve only I = neutrons.
In the case of magnesium the inelastic scattering has been observed
directly by Little, Long, and Mandeville (L20). They scattered 2.5Mev
neutrons (from a DD source) in a block of magnesium and measured
the energy distribution of the scattered neutrons by observing proton
recoils in a cloud chamber. In addition to the elastically scattered
group, they observed a single group of inelastically scattered neutrons
whose energy corresponds to an excited state of magnesium at 1.30 Mev
(compare Table 7a). From the strengths of the two groups, they com
puted values of ~1.6 barns and ~0.6 barn for the elastic and inelastic
scattering cross sections, respectively, in excellent agreement with the
result of Grace et at. and the measured value of c t ~ 2.2 barns. The
value of Ti/T, shown in parentheses in Table 7b, is computed from their
value of <ti « 0.6 barn.
302
The Neutron
[Pt. VII
TABLE 7
(a) Results of Grace, Beghian, Preston, and Halban (B13, G27) on Inelastic
Scattering of 2.5Mev Neutrons, from Observation of the Resulting GammaRadiation
Element
Atomic Weight
Observed
E y (Mev)
Observed
a\ (barns)
Be
9
<0.014
C
12
<0.006
F
19
1.3 ±0.1
0.52 ±0.18
Mg
24(77%), 25(12%),
26(11%)
1.4 ±0.1
0.75 ±0.23
S
32
2.35 ±0.15
0.38±0.1
Cr
52(84%), 53(10%),
50(4%), 54(2%)
1.4 ±0.1
1.2 ±0.4
Fe
56(92%), 54 (6%),
0.8 ±0.1
1.8 ± 1.3
57(2%), 58(0.3%)
2.2 ±0.2
0.14 ±0.05
Cu
63(69%), 65(31%)
1.1 ±0.1
1.2 ±0.6
2.2 ±0.1
0.34 ±0.12
(b) Interpretation
Tx/r
Ele
(barns)
Refer
ence
Estir
<r c (b
nated
arns)
ment
Experii
nent
Theory
F
2.7
W32
1
.2
0.4
0.41
Mg
2.2
G20, A2
1
.5
0.5 (0
.4)
0.40
S
2.8
G20, A2
2
0.2
0.19
(2) Medium Nuclei. In the scattering of fast neutrons by medium
nuclei, the product nucleus can be left in any one of many (energetically)
available excited states; correspondingly, many groups of inelastically
scattered neutrons will be emitted by the compound nucleus. The con
siderations applied in the previous section for a single level can easily be
generalized to the case of many levels. Let c; be the cross section for
the excitation, through inelastic scattering, of the z'th level of the target
nucleus (excitation energy Z>,) :
<n = (42b)
Sec. 2C] Interaction with Nuclei 303
The partial width I\ is obtained by the substitution of the subscript i
for 1 in Eqs. (38a) and (33a).
Since, for intermediate nuclei and fast neutrons, neutron reemission
is usually predominant over all other processes, the total width is
n
r = r + r : +• • •+ i\ + • • •+ r n = £ r, (28b)
in which the first term is the width for elastic reemission and the suc
ceeding terms correspond to inelastic scattering in which all possible
levels (up to the highest for which D n < E) of the target nucleus are
excited.
As in the case of onelevel excitation, the value and energy depend
ence of a specific I\ are determined by the angular momentum proper
ties of the emitted neutrons which, in turn, derive from the spins and
parities of the levels involved. Since, for medium compound nuclei
formed by the capture of fast neutrons, many levels involving many
angular momenta are likely to be excited, decay to a given product
nucleus state will be possible through the emission of neutrons of a
number of different angular momenta. In the ensuing competition,
1 = emission when possible, will usually predominate. However, for
some levels, I = emission will be impossible, and these will usually be
less strongly excited. For a complete description it is, of course, neces
sary to take into account all the possible I values of the emitted neutrons.
Nevertheless, for purposes of illustration it is of interest to consider
the energy dependence of the inelastic scattering cross sections under
the assumption of <Swave scattering only. The values of the first few
<Ti vs. E are shown in Fig. 14; in addition to the assumption of $wave
scattering, we have also assumed uniform level spacing for the product
nucleus, i.e., Z>i = D, D t = iD. The uppermost curve, labeled a oc , is
the relative cross section for capture elastic scattering. At a given
value of the incident neutron energy E, the total cross section for in
elastic scattering is
n
"in = z2 c i ~ °c ~ "oc (68')
i=l
The data on the inelastic scattering of fast neutrons by intermediate
nuclei are meagre. Barschall, Battat, Bright, Graves, Jorgensen and
Manley (B8) have measured the energy distribution of 3.0 and 1.5 Mev
neutrons scattered by iron, using, as an energysensitive detector, pro
ton recoil proportional counters with different "bias" values. Although
the resolution of these measurements was rather crude, the results can
be satisfactorily understood in terms of the theory outlined above and
304
The Neutron
[Pt. VII
the (three) known levels of Fe 56 of excitation energy less than 3 Mev
(F9). The results of Grace et al. (G27), shown in Table 7a, are in fair
agreement with those of Barschall et al. for iron.
(3) Heavy Nuclei. In the scattering of fast neutrons by heavy nuclei
a large number of energy levels of the target nucleus are available for
excitation. Separate groups of inelastically scattered neutrons will, in
general, not be observed, both because of the limited resolving power of
Fig. 14. Cross sections, <r,, for the excitation of the lowlying levels of the target
nucleus as a function of the incident neutron energy, E. The curves are based on
the assumptions: (1) constant level spacing; (2) S scattering only; (3) I\ =
C{E — Ei) Vi , with C constant over all levels.
all fastneutron detectors and because the separation of levels in the
product nucleus is usually smaller than the spread in energy of avail
able fastneutron sources. The inelastically scattered neutrons will
appear to have a continuous spectrum (except, possibly, for the elasti
cally scattered and a few adjacent highenergy groups) ranging from
zero to the incident energy E.
Because of the large number of levels involved, it is no longer fruitful
to attempt to interpret such experiments in terms of a theory involving
the properties of individual levels. Instead, it is possible to apply a
statistical theory, as developed by Weisskopf (W16, B34). The statisti
cal theory predicts an energy distribution of the inelastically scattered
neutrons which is of the Maxwellian form
da(e,E) ^<To— 2 e E/r dz
s
Ji2
(69)
Sec. 2C] Interaction with Nuclei 305
In the above, da(e,E) is the cross section for the scattering of a neutron
of initial energy E into the energy between e and s + de. The constant
T, referred to as the nuclear temperature, is a measure of the excitation
of the product nucleus after the emission of the inelastically scattered
neutron. Strictly speaking, it is not energy independent but rather a
function of the excitation energy of the residual nucleus, E — e. How
ever, for most cases of interest, T « E, and the major part of the spec
trum of inelastically scattered neutrons is in the energy region s <JC E;
for this part of the spectrum, T may be regarded as essentially con
stant, and roughly corresponding to the full possible excitation energy
of the product nucleus. However, for the highenergy portion of the
spectrum, for which e ~ E, Eq. (69) is not a good approximation to
the inelastically scattered neutron energy distribution.
The energy dependence of Eq. (69) can be understood in terms of two
opposing factors in the competition between the various possible modes
of deexcitation of the compound nucleus through neutron reemission:
(1) The energy dependence of the neutron scattering width favors the
emission of highenergy neutrons. This effect is responsible for the first
factor e in Eq. (69) when proper account is taken of the emission of neu
trons in all possible angular momentum states. (If neutrons were emit
ted only in the I = state, the factor would be s H .) (2) The number of
available levels of the product nucleus increases rapidly with the exci
tation energy, thus favoring the emission of lowenergy neutrons. This
effect leads to the exponential factor in Eq. (69).
The competition between the two factors results in a maximum, in
the scattered neutron energy distribution, at an energy intermediate
between and E. T, the temperature of the product nucleus, is a
measure of its level density at the excitation energy remaining after
the emission of the inelastically scattered neutron. From statistical
mechanical considerations, it can be shown that (B34)
1 d
= [In u(E')] (70)
T(E') dE' K J
where w(E') is the nuclear level density at the excitation energy E'
= E — s. For an exponential energy dependence of the nuclear level
density, as given by Eq. (41),
T(E') ~ (J (70a)
In heavy nuclei, the very rapid increase of nuclear level density with
increasing excitation energy has the effect that the maximum of the
306
The Neutron
[Pt. VII
scattered neutron energy distribution is at relatively low energies; cor
respondingly, the excitation energy of the product nucleus is, for the
major fraction of the inelastic scattering, at an energy close to the maxi
mum possible excitation energy, E' <~ E. [This rapid increase in level
density corresponds to large values of the constant a (Table 4) and
2.4
2.0
1.6
1.2
0.8
i 0.4
w
(a) E
= 3 Mev
^ ■
1
I
,
1.0
1.6
E(Mev)
 /
^v (b).E=1.5 Mev
I

^ ~ 1
0.5
E(Mev)
1.0
Fig. 15. Energy distribution of inelastically scattered neutrons from wolfram.
The experimental results of Barschall et al. (B8) are plotted as histograms. The
smooth curves are derived from the statistical theory and Eq. (69) (F9).
hence to small values of the nuclear temperature, T <K E.] Thus, over
the largest part of the spectrum, the nuclear temperature can be closely
approximated by
T(E)
0"
(70b)
which is independent of the energy of the scattered neutrons. It is in
this approximation that Eq. (69) is valid.
The observations of Barschall et al. (B8), on the inelastic scattering
of 1.5 and 3.0 Mev neutrons by wolfram (tungsten), can be interpreted
in terms of the statistical theory and Eq. (69) (F9). The results of their
measurements on the energy distribution of the inelastically scattered
neutrons are plotted as histograms in Fig. 15, together with the predic
tions of the theory (smooth curves). The theoretical curves have been
% (E,E t ) = C
Jo
Sec. 2C] Interaction with Nuclei 307
computed on the assumption of T = 0.5 and 0.35 Mev for E = 3.0 and
1.5 Mev, respectively, corresponding to a = 12 Mev 1 , in good agree
ment with Table 4.
Table 8 is a collection of integral inelastic scattering cross sections
and temperatures for medium and heavy nuclei. The measurements
are represented by the cross section for the scattering of a neutron of
incident energy E to an energy e < E t ,
dc(E,z) (69a)
'o
and by the temperature corresponding to the emitted neutron distribu
tion. Only those measurements have been included for which the inci
dent neutron energy and the detector threshold are relatively well
defined. Some of these are amenable to interpretation along the lines
outlined above in the discussion of the results for wolfram. For most,
the energy sensitivity of the threshold detector used is not sufficiently
well known to permit more quantitative conclusions.
However, these measurements do give an indication of the energy
distribution of the inelastically scattered neutrons. In particular, they
show that, also with regard to inelastic scattering, the magic number
nuclei lead and bismuth behave in an anomalous fashion; their scatter
ing is much more similar to that of medium nuclei than to that of heavy
nuclei. The importance of this observation is that it relates to the rela
tively lowlying level spacings of the target nuclei, which appear, from
these observations, to be anomalously large for lead and bismuth. The
previously mentioned evidence on the small capture cross sections and
absence of slowneutron resonances for these nuclei reflect a wide level
spacing of the compound nucleus.
In addition to the direct observation of inelastically scattered neutrons
or the resulting gammaradiation, there is another method of detecting
inelastic scattering, applicable only to certain special nuclei. These are
nuclei which have a metastable (long halflife) level at an energy less
than that of the incident neutrons; when excited, the metastable state
can be detected by the resultant radioactivity. The metastable state
of In 115 (energy 340 kev, halflife 4.5 hr) was first observed by Gold
haber, Hill, and Szilard (G13) as resulting from the inelastic scattering
of fast neutrons.
A metastable state can be induced either by direct excitation (the
product nucleus is left in the metastable state), or indirectly, through
the excitation, by inelastic scattering, of a higher state of the product
nucleus, which subsequently decays (by gammaray emission) to the
metastable state. In the first case, Eqs. (42a) and (38a) determine the
308
The Neutron
[Pt. VII
TABLE 8
(a) Measured Values of Integral Inelastic Scattering Cross Sections, <n n {E,E t )
Ele
ment
E (Mev)
E t (Mev)
Detec
tor t
<Ti n (E,E t ) X
(barns)
Refer
ence
<j c (barns)
(Fig. 10)
Be
2.5
2.5
y
<0.014
G27
14
~3
Al
0.16 ±0.07
P15
~11
Cu
0.82 ±0.03
P15
0.79
B
14
~3
Al
0.24 ±0.04
P15
~11
Cu
0.69 ±0.10
P15
0.64
C
2.5
2.5
7
<0.006
B13
14
~11
Cu
0.85 ±0.02
P15
0.74
F
2.5
2.5
7
0.52 ±0.18
G27
1.4
Mg
2.5
2.5
7
0.75 ±0.23
G27
1.10
~2.5
cc
~0.6
L20
Al
14
~3
Al
0.62 ±0.07
P15
~11
Cu
1.06 ±0.05
P15
0.93
S
2.5
2.5
7
0.38±0.1
G27
0.90
Cr
2.5
2.5
7
1.2 ±0.4
G27
1.47
Fe
1.5
~0.5
pc
B8
~0.9
pc
0.6
B8
1.58
2.5
2.5
7
1.9 ± 1.3
G27
1.52
3.0
~0.75
pc
0.3
B8
~1.50
pc
0.7
B8
~2.25
pc
1.1
B8
1.50
14
~2
p
0.78 ±0.03
P15
~3
Al
1.21 ±0.03
P15
~11
Cu
1.45 ±0.02
P15
1.34
Co
1.5
~0.5
pc
(0)
B8
~0.9
pc
(0.2)
B8
~1.3
pc
(0.8)
B8
1.62
Ni
1.5
~0.5
pc
(0)
B8
~0.9
pc
(0.1)
B8
~1.3
pc
(0.6)
B8
1.62
Sec. 2C]
Interaction with Nuclei
309
TABLE 8 (Continued)
(a) Measured Values of Integral Inelastic Scattering Cross Sections, <Ti n (E,E t )
Ele
ment
E (Mev)
E, (Mev)
Detec
tor f
a in (E,E t ) t
(barns)
Refer
ence
(T c (barns)
(Fig. 10)
Cu
1.5
~0.5
pc
(0.3)
B8
~0.9
pc
(0.6)
B8
~1.3
pc
(0.9)
B8
1.65
2.5
2.5
y
1.5 ±0.7
G27
1.62
3.0
~0.75
pc
(0.6)
B8
~1.50
pc
(1.3)
B8
~2.25
pc
(1.5)
B8
1.61
Cd
14
~2
p
1.14 ±0.04
P15
~3
Al
1.66 ±0.07
P15
~11
Cu
1.89 ±0.06
P15
1.85
Ta
1.5
~0.5
pc
(1.4)
B8
~0.9
pc
(2.0)
B8
~1.3
pc
(2.7)
B8
2.57
W
1.5
~0.5
pc
0.9
B8
~0.9
pc
2.1
B8
2.57
3.0
~0.75
pc
1.4
B8
~1.50
pc
2.4
B8
~2.25
pc
2.8
B8
2.63
Au
3.0
~0.75
pc
(2.1)
B8
~1.50
pc
(2.8)
B8
~2.25
pc
(3.0)
B8
2.69
14
~2
p
1.47±0.10
P15
~3
Al
2.06 ±0.09
P15
~11
Cu
2.51 ±0.04
P15
2.69
Pb
1.5
~0.5
pc
B8
~0.9
pc
0.4
B8
2.81
2.5
~1
U
0.55
S65
~2.5
cc
1.3 ±0.5
D17
2.77
3.0
~0.75
pc
0.7
B8
~1.50
pc
1.2
B8
~2.25
pc
1.6
B8
2.76
14
~2
p
0.91 ±0.06
P15
~3
Al
2.29 ±0.04
P15
—11
Cu
2.56 ±0.05
P15
12
PP
<2.6
W21
2.77
14.5
~3
Al
2.20 ±0.17
G6
~11
Cu
2.29 ±0.12
G6
2.76
310
The Neutron
[Pt. VII
TABLE 8 (Continued)
(a) Measured Values of Integral Inelastic Scattering Cross Sections, Oin(E,E t )
Ele
ment
E (Mev)
E t (Mev)
Detec
tor f
Oin(E,Et) %
(barns)
Refer
ence
<r c (barns)
(Fig. 10)
Bi
2.5
~1
U
0.64
S65
2.77
14
~2
P
1.03 ±0.11
P15
~3
Al
2.28 ±0.08
P15
~11
Cu
2.56 ±0.05
P15
12
VP
<3.3
W21
2.78
(b) Temperatures for Observed Distributions of Inelastically Scattered Neutrons
(Eq. 69)
Element
E (Mev)
Range of
e (Mev)
T{E) (Mev)
Reference
B
11
>4
2.3±0.3
G31§
14
0.9±0.1
G31
Al
15
>1
1.1 ±0.1
S53
Si
10.6
>2
1.3±0.1
G31
Fe
1.5
<0.9
none 
B8, F9
3.0
<2.25
none
B8, F9
15
>1
0.6±0.1
S53
Co
10.5
>2
0.95 ±0.1
G31
Pd
14
>2
0.85±0.1
G31
W
1.5
<0.9
0.35
B8, F9
3.0
<2.25
0.50
B8, F9
Au
3.0
<2.25
(0.33)
B8, F9
Hg
14.6
>2
0.8± 0.1
G31
Pb
1.5
<0.9
none
B8, F9
2.5
>1
none
D17, F9
3.0
<2.25
none
B8, F9
4.3
>1
none
M2
Sec. 2C]
Interaction with Nuclei
311
TABLE 8 {Continued)
(b) Temperatures for Observed Distributions of Inelastically Scattered Neutrons
{Eq. 69)
Element
E (Mev)
Range of
e (Mev)
T{E) (Mev)
Reference
14
13
—0.8
W21
14.3
>2
0.78±0.1
G31
15
>1
0.7±0.1
S53
Ki
4.3
>1
none
M2
14
15
—0.9
W21
f The symbols have the following meanings :
7 = direct detection of gammarays from the product nucleus
P = T 31 {n,p) reaction
Al = Al 27 (n,p) reaction
Cu = Cu 63 (ra,2w.) reaction
pc = detection in a biased proton recoil proportional counter
pp = detection by proton recoils in a photographic emulsion
cc = detection by proton recoils in a cloud chamber
U = detection by U 238 fission in an ionization chamber
t Values in parentheses have not been corrected by the authors (B8) for
effects of multiple scattering in the target.
§ Reference (G31) is to measurements of Gugelot on neutron distributions
from {p,n) reactions. In these cases, the element given is that of the product
nucleus and the energy (second column) is the maximum neutron energy, i.e.,
the proton kinetic energy plus the reaction Q value.
 "None" means that the inelastic scattering cannot be treated by the statis
tical theory. Instead, individual levels must be considered.
behavior of the cross section for the excitation of the metastable level;
in the second case, the partial cross section for the production of the
metastable state through the excitation of the ith level is given by the
product of Eq. (42a) and a factor representing the relative probability
for the decay of the ith level to the metastable state. The excitation
function (total cross section vs. neutron energy) is the sum of the partial
cross sections. It will exhibit discontinuities at the energies correspond
ing to the inception of excitation of those product nucleus levels which
have an appreciable probability of decaying to the metastable level.
The cross section for the excitation of the metastable state of In 115
by neutrons of energy up to —4 Mev has been measured by Cohen
312 The Neutron [Pt. VII
(CI 7). The resolution of his measurements was too crude to detect the
effects of individual levels of the product nucleus. More recently Ebel
(El) has measured the excitation function for In 115 * with good resolu
tion. He observes a threshold at 600 kev (no direct excitation of the
metastable level) and discontinuities corresponding to additional prod
uct nucleus levels at 960 and 1370 kev. Ebel has also measured the
cross section for the excitation of the 540kev metastable level (halflife
7 sec) of Au 197 . This level can be directly excited, and also excited
through levels at 1.14 and 1.44 Mev.
Since the metastable state is characterized by a large difference in
spin from the ground state, the levels of the product nucleus which are
involved in its excitation are, perforce, those whose spins differ appre
ciably from that of the ground state. The shape and magnitude of the
cross section for excitation of the metastable state can be used to deter
mine (within limits) the spins and parities of the states involved in its
excitation (El).
II. Diffraction or Shadow (NonCapture) Elastic Scattering
In the previous discussions, we have considered the total cross section
as consisting of two parts : (1) that portion which leads to the formation
of a compound nucleus, <r c , and (2) that portion corresponding to proc
esses in which the incident neutron merely changes its direction, with
out ever effecting a change in the quantum state of the target nucleus; l
this elastic scattering, <r e i (see Eq. 68), is variously referred to as diffrac
tion or shadow scattering.
For slow and intermediate neutrons (X > R) capture and diffraction
scattering are intimately connected because of the interference between
resonance and potential scattering. For fast neutrons (X < R) the two
processes separate in a natural fashion. In the first place, the position
(extent) of the neutron is relatively well defined. Those neutrons which
strike the nucleus have a high probability of penetrating its surface,
whereupon they are captured into a compound nucleus. While a cer
tain fraction (relatively large for light nuclei, small for medium and
heavy nuclei) may be reemitted with the full energy, these (capture)
elastically scattered neutrons can, except for interference effects in the
1 The target nucleus, initially at rest, will of course receive momentum and kinetic
energy as a result of the scattering, and the neutron will, correspondingly, lose energy.
However, by the term "quantum state" we refer to the state of internal motion of
the target nucleus, which is unaffected unless the incident neutron penetrates the
nuclear surface.
Sec. 2C] Interaction with Nuclei 313
neighborhood of a resonance, be separated from the diffraction scattered
neutrons by virtue of their different angular distribution.
Shadow scattering, on the other hand, results from the diffraction of
those neutrons which pass close by, but not into, the nucleus. Thus,
despite the fact that we are considering the energy range for which
X < R, in which neutrons may be expected to exhibit a minimum of
wave properties, it is precisely the wave nature of the incident neutron
beam which leads to the phenomenon of shadow scattering. Further
more, while the angular distribution of the diffraction elastically scat
tered neutrons depends on the value of the nuclear radius (specifically,
on kR = R/X), the nature of the diffraction scattering process is inde
pendent of the nuclear atomic number.
To a good approximation, shadow scattering is analogous to the
diffraction of a plane wave (say light) by a spherical obstacle, for wave
lengths small compared to the size of the obstacle. The scattered neu
trons are confined to within a relatively small angle,
0o « ^ (71)
An expression for the angular distribution of the scattered neutrons was
first derived by Placzek and Bethe (PI 7) on the basis of the optical
analogy. Recently, Hauser and Feshbach (H44) have derived a more
accurate formula,
~^ = ' cot 2 (){Ji[fc(fl + X) sine]} 2 (72)
dQ. 4 \2/
for the cross section per unit solid angle for scattering of neutrons into
the angle 9; J t is the Bessel function of the first kind. The cross section
has a maximum at = 0, falls to zero at k{R + X) sin 6 = 3.83, and then
goes through a series of subsidiary maxima and minima.
For very small wavelengths, X <K R, the cross section for shadow
(elastic) scattering is
"el
= fdv(0) « r(B + X) 2 (73)
At this extreme, a t « 2ir(R + X) 2 (Eq. 67), so that diffraction elastic
scattering accounts for approximately half of the total cross section for
fast, very fast, and ultra fast neutrons.
The experiments of Amaldi, Bocciarelli, Cacciapuoti, and Trabacchi
(A21) on the angular distribution of the elastic scattering of 14Mev
314
1800
1700
1600
1500
1400
1300
1200
1100
1000
900
800
700
600
500
400
300
200
100
The Neutron
[Pt. Vii
v?=
8.5
a(8;
vs. 6
_
E
=14 W
ev
R='t
.5 I
^w \
to
R = (
5.5
R = l
>.5
R=A
??~~~
X
X
x
X
X
i? = 3.5
10 15 20 25 30 35 40 45 50 55 60 65
Fig. 16. Angular distribution in diffraction elastic scattering of 14Mev neutrons
for a number of nuclear radii (given in units of 10~ 13 cm). The curves are accord
ing to the theory of Hauser and Feshbach (H44). The crosses are from the measure
ments of Amaldi and coworkers on lead (A21).
Sec. 2C] Interaction with Nuclei 315
neutrons by lead are in excellent agreement with the theory. In addi
tion to the general shape of the scattered neutron distribution, they
have observed subsidiary maxima at approximately the expected angles.
The theoretical predictions for E = 14 Mev and a number of values of
the nuclear radius are shown in Fig. 16. Also plotted are the experimen
tal points of Amaldi and coworkers (A21).
III. Angular Distribution of Scattered Neutrons; the
Transport Cross Section o>
The angular distribution of the scattered neutrons, with respect to
the direction of the incident neutrons, is usually described in terms of a
differential scattering cross section, <r sc (0), the cross section for scatter
ing into unit solid angle at the angle 6, in the laboratory system,
f«c
= f<Tac(6) da (73a)
Deviations from spherical symmetry result from a number of possible
causes, some of which have previously been discussed. Since the angu
lar distributions are determined by the nature of the scattering process,
measurement of o sc (0) can sometimes lead to useful nuclear information.
The angular distribution of scattered neutrons is also important in
determining the rate of diffusion of neutrons through matter (to be dis
cussed in a subsequent section). In such problems, the important quan
tity is the transport cross section,
atr = <rt J <T, C (0) cos 6 da (74)
a tr determines the rate at which the neutron loses its forward momentum
(or the memory of its original direction). Under certain special assump
tions concerning the nature of the scattering process simple expressions
for <Tt r can be derived.
(1) Light Nuclei. In the scattering of fast neutrons by light nuclei,
elastic scattering is the most important process. Furthermore, the
scattering is (except at the highest energies) mainly £wave scattering,
i.e., spherically symmetrical in the centerofmass coordinate system
(c.m.s.). Neglecting all processes but $wave elastic scattering, the
angular distribution of the scattered neutrons is still not symmetrical in
the laboratory system (l.s.), owing to the forward motion of the center
of mass.
316 The Neutron [Pt. VII
Thus, for a neutron, scattered through the angle <p, in the c.m.s., by
a nucleus of atomic weight A, the angle in the l.s. is given by
(1 + A cos <p)
(A 2 + 1 + 24 cos?)'
cos 6 = , A 2 , , , »„ — VA W
E (75a)
The energy in the l.s. of the elastically scattered neutron, of initial
energy E, is
r(A 2 + l + 2Acos**r
E =
L {A + l) 2
the difference (E — E') going to the recoil nucleus. The energy of the
scattered neutron varies between a maximum, E mB ,^' = E, for <p = 8
= 0, and a minimum,
_ (A  iy 2
Er,
■'[
(A + 1).
# b a E (75b)
for ? = 180°. For <Swave scattering (in the c.m.s.) the scattered neu
trons are distributed uniformly in energy between E and aE. Corre
spondingly, under the above assumptions,
2
(Ttr = Ot ~ — "el (74a)
oA.
(2) Medium and Heavy Nuclei. It is evident from Eq. (74a) that
the effect of the motion of the c.m.s. can be neglected for medium and
heavy nuclei. On the other hand, elastic scattering is no longer spheri
cally symmetrical, owing to diffraction effects. Most of the capture
leads to inelastic scattering, and the resulting neutrons are not necessarily
symmetrically distributed, especially if only a few levels are involved
(medium nuclei). However, for most purposes, it is reasonable to treat
the inelastic scattering as though it were spherically symmetrical, and
to substitute a e i for u sc in Eq. (74). With this assumption the theory
of Feshbach and Weisskopf (F28) can be used to predict the energy de
pendence of atr The results are shown in Fig. 17, together with the
experimental values of Barschall et al. (B8).
(d) Very Fast Neutrons. So far as neutron scattering is concerned,
the very fastneutron energy range does not differ in any significant re
spect from the fastneutron range. Most of the preceding discussion
applies equally well to very fast neutrons; indeed, many of the experi
mental results quoted were for neutrons of energy > 10 Mev. However,
in the extension of the previous considerations into the very fastneutron
region, the following points must be kept in mind: (1) Since many levels
of the target nucleus can be excited by very fast neutrons, except for a
Sec. 2C]
Interaction witrfNuclei
317
few of the lightest nuclei, the inelastic scattering must be treated accord
ing to the statistical theory. (2) Because of the large excitation energy,
the compound nucleus has, in addition to neutron reemission, other
possible modes of decay. Thus, for light and some medium nuclei,
(n,p), (n,a), etc., reactions compete favorably with neutron reemission.
For medium and heavy nuclei, these reactions are also possible, but the
{n,2ri) reaction is the most probable competing process. Although we
postpone the detailed discussion of this reaction, it is important to note
N\^o=
>
+ Fe
o W
A Pb
Xo=8 \N
. +
Ir'^^^l
■ i
\
A ' : =:==.
i i i i
i i I i
i i i i
i i i i
Fig. 17. a tr vs. x = kR for nuclei of X = KoB = 5 and 8; according to Feshbach
and Weisskopf (F28). The points, for iron, wolfram, and lead, are from the measure
ments of Barschall et al. (B8).
that its presence has a strong effect on the observed energy distribution
of the emerging neutrons. Experiments designed to measure the energy
distribution of inelastically scattered neutrons must, in the very fast
neutron range, take into account the deviations from Eq. (69) brought
about by the presence of the (n,2n) reaction (S53).
The upper end of the very fastneutron range is characterized, for all
nuclei, by X « R. Thus at these energies <x t ~ 2x(fi + X) 2 and c c «
cr e i » tt{R + X) 2 .
(e) Ultrafast Neutrons. In the ultrafastenergy region, the character
of neutron reactions undergoes a gradual change. The simple picture
of neutron capture into a compound nucleus, as developed in the previ
ous sections, is no longer applicable. Instead, as pointed out by Serber
(S21), the nucleons in the nucleus must be treated as relatively inde
pendent particles, since their energy of internal motion and of binding is
smaller than the energy of the bombarding neutron. Furthermore,
owing to the rapid decrease of the primary (np) and (nn) cross sections
with increasing neutron energy — roughly as 1/E in the energy range 50
318 The Neutron [Pt. VII
to 250 Mev — the incident neutron has an appreciable probability of
passing through the nucleus without undergoing any collisions, i.e., the
nucleus is partially transparent to ultrafast neutrons, the more so the
lighter the nucleus.
Under these circumstances the probability of a nuclear reaction can be
described in terms of an absorption coefficient (reciprocal mean free
path) of nuclear matter for neutrons,
3Ao
* = ^ Cm (76)
where
Za{np) + (A  Z)a(nn)
a =  (76a)
A
[In Eq. (76a) a(np) and <r(nn) are appropriate averages over the spec
trum of relative energies of the incident neutron and the nucleons in the
nucleus, appropriately reduced to take into account the effect, due to
the exclusion principle, that not all energies are available to the recoiling
nucleons.] Defining <r„ as the cross section for a collision of the incident
neutron inside the nucleus, integration over a spherical nucleus of radius
R yields
,1 1  (1 + 2 K R)e 2KR }
The nuclear transparency also influences the diffraction elastic scat
tering of ultrafast neutrons, both in the magnitude of <?d — <r e i and in
the angular distribution of the scattered neutrons. These effects have
been considered in the calculations of Fernbach, Serber, and Taylor
(F25). They have derived the diffraction scattering of a partially
transparent sphere, characterized by the propagation vector k + fci
= [2m(E + V)] 1A /h. (V is the effective nucleon potential inside the
nucleus; k = 1/X is the incident neutron wave number.) The results
of these calculations, for the special case kyJK = 1.5, are shown in Fig. 18,
in which a^, c a , and cr t = a a + <?d are plotted against kR. For large
values of kR (complete opacity) <r a » ad — > kR 2 .
Fernbach, Serber, and Taylor have compared their calculated values
of (T t with the measurements of Cook, McMillan, Peterson, and Sewell
(C25), for E « 85 Mev. They find excellent agreement, assuming R
= 1.37A M X 10~ 13 cm, k = 2.2 X 10 12 cm 1 , and k t = 3.3 X 10 12 cm 1
(V = 30.8 Mev). The measurements of <r t at E « 95 Mev, by DeJuren
and Knable (D9) are also in good agreement with theory. These investi
gators have, in addition, determined a a for two elements (carbon and
copper), by a "poor geometry" attenuation measurement, and obtained
values in good agreement with the theory.
Sec. 2C]
Interaction with Nuclei
319
The angular distributions of 84Mev neutrons elastically scattered by
aluminum, copper, and lead have been measured by Bratenahl, Fern
bach, Hildebrand, Leith, and Moyer (B59). The results are shown in
Fig. 19 in which the dotted curves are the predictions of the theory of
Fernbach, Serber, and Taylor.
Measurements of the total cross sections of a number of elements have
been carried out at 280 Mev by Fox, Leith, Wouters, and MacKenzie
(F41), and at 270 Mev by DeJuren (D10), who also obtained lower
Fig. 18. Absorption, diffraction, and total cross sections as functions of the nuclear
radius, measured in mean free paths for the incident neutron. The curves are for
ki/i< = 1.5, corresponding to neutrons of E « 90 Mev (F25).
limits for <r of carbon, copper, and lead by "poor geometry" attenuation
measurements. The abovementioned measurements of a t are included
in the compilation of Adair (A2). Cross sections of some representative
elements have been measured at a number of intermediate energies —
between 110 and 240 Mev— by DeJuren and Moyer (D12). A rather
interesting common feature of all the observed <x t vs. E curves is their
comparative independence on energy above <~160 Mev. Another point
of interest is the relative constancy of the ratio <r t (270 Mev)/<r* (95
Mev) ; its value is 0.57 between beryllium and tin and then rises slowly
to 0.67 for uranium.
The last feature is difficult to reconcile with the partially transparent
nucleus model, which predicts considerably greater nuclear opacities
for the heavier nuclei. 1 Indeed, for the 270Mev data, it is only possible
1 This peculiar behavior of <r ( for neutron energies > 200 Mev may possibly be
associated with the inception of meson production. While the threshold for pion
creation is ~275 Mev in a nucleonnucleon collision, meson production is observed
at considerably lower energies for the bombardment of nuclei by nucleons. This is
due to the internal motion of the nucleons in the nucleus.
320
The Neutron
[Pt. VII
o 1 
1
1 '
Al
— r — ' — i — ' I ' I ' i '
1
1, 
o Carbon detector measurement
~^
1
X*
1 .
1,1,1,1 .~T^
24°
20
.... , ,
1 Cu
rT — ' "7" i 1 I 1 ' 1 »
18
~
c
a is
1
"
o
,4
S 14
to
 ii c
"^0
""
a &
\
VI
1 1°


\
\
a
\
\
4
\
2
'
. I.I, r« J. la—
0°
12° 16° 20°
80
ft 70
.2
% 60
w
at
O 50
O
1 40
0)
S 30
— i — i — i — i 1 — i — r— i —
 Pb
,,.. , _, ,

Ntf
•
\
\
\
\
1 i 1 ^tf^eT—
TTJ
'
4° 8° 12° 16° 20° 24°
Neutron scattering angle
Fig. 19. Differentia] cross sections for elastic scattering of 84Mev neutrons by
aluminum, copper, and lead, due to Bratenahl, Fernbach, Hildebrand, Leith, and
Moyer (B59). The curves are the predictions of the theory of Fernbach, Serber,
and Taylor (F25).
Sec. 2C] Interaction with Nuclei 321
"to obtain a reasonable fit . . . [if] the potential change experienced by
the bombarding neutron when entering a nucleus [is] dropped to zero"
(D10).
2. Radiative Capture (n,y). The excited compound nucleus, produced
by neutron capture, can decay to its ground state through the emission
of one or more gammarays. The (n,y) reaction competes with neutron
reemission (always possible) and with any other mode of decay which
is energetically permitted. The relative probability of the (n,y) reac
tion is determined by the value of the gammaray width, r r , as com
pared to the widths for all other possible modes of decay:
*(n,y) =  1 (42')
At the excitations corresponding to neutron binding energies, the
emission of a gammaray takes place in a time of ~10~ 14 sec; corre
spondingly (Eq. 27) r T ~ 0.1 ev. This value is relatively independent
of the atomic weight of the nucleus involved and of the incident neutron
energy (at least for slow and intermediate neutrons), although con
siderable fluctuations, from nucleus to nucleus and from level to level
in the same compound nucleus, are to be expected. The above lifetime
refers, of course, only to the emission of the first gammaray, after
which the compound nucleus is usually committed to radiative decay,
not having sufficient residual energy to decay by any other mode. In
some cases, successive gammaray emission leads to a metastable (iso
meric) state of the compound nucleus. Usually, however, the gammaray
emission leads, either directly or through a cascading descent, to the
ground state of the compound nucleus.
The study of the energy distribution of the capture gammaradiation
is of considerable interest, since it yields information on the level struc
ture of the compound nucleus, on the nature of the gammaray emission
process, and on neutron binding energies. Furthermore, knowledge of
the capture gammaray spectrum is of practical importance in the prob
lem of shielding neutron chain reactors. However, since further discus
sion of these problems at this point would lead us too far afield, we shall
be content merely to point out that the observation of the capture
gammaradiation is one means of detecting the (n,y) reaction.
In many (n,y) processes the product nucleus is a stable isotope of the
target, e.g., Cd 113 (n, 7 )Cd 114 ; in such cases, the capture gammarays
are the only reaction products. However, for most nuclei, radiative
capture leads to an unstable (radioactive) product nucleus. The reac
322 The Neutron [Pt. VII
tion can then be detected and identified by observing the resulting radio
activity. Since these radioactive nuclei are most likely to have an excess
of neutrons, they usually decay by negative betaray emission. There
are a few cases, e.g., Cu 63 (n,7)Cu 64 , in which the product nucleus can
decay by positron emission or K capture.
Most of the factors which determine the energy dependence of <r(n,y)
have previously been described, both in the general discussion of the
cross section for compound nucleus formation, a c , and in Section 2C1
on neutron scattering. In the following, we summarize the main fea
tures of the radiative capture process.
(a) Slow Neutrons. The value of <r(n,y) at a given slowneutron en
ergy is determined by two factors:
(1) The position and characteristics of the closest resonance or, if the
energy is far from any one resonance ( E — E„ \ ~2> r,), the positions
of the closest resonances and their possible interference effects at the
energy under consideration. The last aspect has been discussed by
Wigner and coworkers (W29, W30, T5). The behavior of <r{n,y) in the
vicinity of a resonance has been considered in detail (Section 2B3, espe
cially Eqs. 52, 56, 58, and 59).
(2) The relative value of the gammaray width, r 7 , to the total width,
r = T y + V n + T a + • • • ; in all but a few light nuclei only the first
two terms, in the above expression for r, differ from zero. Since T y is,
as per the previous discussion, essentially fixed, the competition between
radiative capture and neutron scattering is primarily determined by the
value of r n , which, in turn, is a function of the neutron energy and the
compound nucleus level spacing, according to Eqs. (37) and (38).
From these considerations it is clear that the importance and the
character of the slow neutron (n,y) cross section depend on the atomic
weight of the target nucleus. In light, medium, and some heavy nuclei,
levels are widely spaced, T n >?> T y , and radiative capture plays a minor
role. In most heavy nuclei, on the other hand, levels are closely spaced,
r 7 2> r„, and radiative capture is the predominant resonance reaction.
The properties of a number of slowneutron resonances are summarized
in Table 9. The values of the resonance energy and level widths are
given, as well as the deduced values of the level spacings, derived from
Eq. (37a), for comparison with the observed (average) spacing between
levels.
Of particular interest in the study and use (to produce artificially
radioactive nuclei, for instance) of radiative capture is the socalled
thermal neutron capture cross section, <r t h.{n,y). This cross section has
been variously defined, sometimes in a not very precise fashion. We
Sec. 2C]
Interaction with Nuclei
323
TABLE 9
Properties of Some SlowNeutron Resonances — Mainly Capture f
Target
Nucleus
E r (ev)
I\ (ev) t
/r„(10 3 ev)§
D* (ev) 
Dobs (ev)
17CF
75
0.3
1.64 X 10 s
2,200
26 Mn 8B
345
0.2
12 X 10 3
7,700
10,000
27 Co 59
123
0.3
2.4 X 10 3
2,800
10,000
3oZn
520
525 X 10 s
28,000
>500
3iGa< 69 ' 71 >
98
(0.3)
130
90
<100
32Ge
95
3900
2600
33AS"
46
0.05
110
120
50
35 Br«5,87)
36
(0.1)
35
86
30
42M0 86
46
(0.1)
450
1,000
50
4BRh 103
1.26
0.20
0.45
6
4 6 Pd 108
25
0.14
~40
~60
>10
47Ag 107
15.9
~0.11
~5
~10
~20
4rAg™
5.17
0.16
8.2
34
—20
4 8 Cd 1W
0.176
0.115
0.46
10
~25
49 In 115
1.45
3.86
9
0.08
(0.08)
(0.08)
1.2
0.2
0.5
14
2
5
6
6
6
61 g b (121, 12 3)
5.8
15
(0.1)
(0.1)
~0.3
~2
~2
~8
7
7
B2 Te 123
2.2
(0.1)
8
70
~1
53I 127
19.4
0.45
0.38
1.2
15
6 2 Sm» 9
0.096
0.074
0.31
13
2
324
The Neutron
[Pt. VII
TABLE 9 {Continued)
Propebties of Some SlowNeutron Resonances — Mainly Capture f
Target
Nucleus
E r (ev)
T y (ev) J
/r„(10 3 ev)§
D* (ev) 
Dobs (ev)
63 Eu 161
0.011
0.081
0.004
0.5
~3
6 3 Eu lB3
0.47
0.20
0.9
18
~3
64 Gd 167
0.028
0.12
0.4
30
66 Dy< 161  163)
1.74
(0.1)
3
30
~3
7 2 Hf m
1.08
2.34
0.12
0.16
0.9
2.8
12
26
2
2
72 Hf 178
7.6
(0.1)
56
150
7 3 Ta 181
6.1
10.3
13.6
20
0.2
(0.1)
(0.1)
0.0007
0.85
0.45
0.7
0.04
3.8
1.8
2.2
5
5
5
5
74W 182
4.15
0.07
1.1
4
~20
74 W 183
7.8
(0.1)
1.0
5.3
20
74 W(184)
~200
(0.1)
~400
~200
~100
74W 186
19.25
0.15
250
400
~150
76 Re 185
4.4
11
(0.1)
0.3
3.5
2
15
5
5
76 Re 187
2.15
5.9
7.2
0.14
0.43
0.09
0.40
4.2
0.5
2.1
3
3
3
76 0s< 189 >
6.5
8.8
(0.1)
(0.1)
1.6
7.5
9
36
6
6
77 I r (Wl. W3)
0.64
0.10
0.2
4
2
77 Ir 193
1.27
~0.15
0.4
5
2
Sec. 2C]
Interaction with Nuclei
325
TABLE 9 (.Continued)
Properties op Some SlowNeutron Resonances — Mainly Capture f
Target
Nucleus
E T (ev)
T y (ev) i
/r B (io 3 ev)§
D* (ev) !
Dobs (ev)
78 Pt< 196 >
11.5
18.2
(0.1)
(0.1)
6.5
5.5
27
19
10
10
79 Au I9r
4.87
0.15
21
136
~50
8oHg (199,201)
2.0
35.5
(0.1)
(0.1)
25
80
250
190
~15
~15
92U 238
~11
~0.20
~8.6
~20
t This table represents a complete reevaluation of the data available in the
open literature up to the summer of 1952. The measurements, upon which the
constants are based, are so numerous that we have not attempted to include
references to them in the table. Instead, the reader is referred to previous com
pilations, of which the most complete are those of Blatt and Weisskopf (B34),
of Wigner (W31), and of Teichmann and Wigner [Phys. Rev., 87, 123 (1952)].
Special attention is called to the most recent neutron cross section compilation
prepared by the AEC Neutron Cross Section Advisory Group and issued by
the U. S. Department of Commerce, Office of Technical Services, as document
AECU2040 (May 15, 1952). We gratefully acknowledge private communica
tions of unpublished results by the Harwell timeofflight velocity selector
group (M26) and by the Argonne fast chopper group (whose members include
L. M. Bollinger, R. R. Palmer, and S. P. Harris). Other data, pertaining to
scattering resonances, can be found in Table 6.
% The values enclosed in parentheses have been assumed in order to compute
/T„ from the measured value of <r T 2 .
§ The numbers quoted are at the resonance energy, except for the negative
energy resonances, where the neutron width is for the energy E = \E r \.
 D* is computed from Eq. (37a): D* = TrK T n /2k. When the statistical
factor / is not known, it is assumed to be 1 for eveneven target nuclei and J
for oddA target nuclei.
326 The Neutron [Pt. VII
shall define it as the average, over a neutron flux with a Maxwellian
energy distribution (Eq. 3), of the slow neutron (n,y) cross section,
,E'
a(n,y) d4(E,E )
<rth(n,y) = ^ (78)
d<f>(E,E )
The cutoff energy, E', is chosen to be sufficiently large so that only a
very small fraction of the Maxwell distribution is at energies E > E'.
(Most frequently, the cadmium cutoff, E' ~ 0.30.5 ev is used; for E
= "Fir ev > less than 10 4 of the neutrons have E > W.) The cutoff is
introduced for purely practical reasons, since (1) the cross sections above
~1 ev, which frequently show many resonances, are not well known,
and (2) most methods for producing thermal neutrons give spectra
which, above ~1 ev, have a 1/E "tail" superimposed on the Maxwell
distribution. Unless otherwise specified, thermal neutron cross sections
are taken to correspond to the Maxwell distribution at a temperature
of ~300°K (E a = jV ev, v Q = 2.2 X 10 5 cm/sec).
If <r(n,y) is a known function of E, a t h can be computed in a straight
forward fashion. For nuclei in which the first resonance is relatively far
from thermal energies, the (n,y) cross section obeys the l/v law,
(59a)
and
(79)
The thermal neutron (n,y) cross section may deviate from the l/v law
in a spectacular fashion if a resonance happens to fall in the thermal re
gion. The cross section of cadmium, shown in Fig. 20a, is a case in
point. However, even if the closest resonance does not fall in the ther
mal region, it may still cause serious deviations from a l/v behavior for
thermal neutrons. The cross sections of iridium (Fig. 7d) and of mer
cury (Fig. 20b) show, respectively, the effects of a close positiveenergy
(E r > 0) and negativeenergy (E r < 0) resonance. Because of such
effects, considerable care must be taken in interpreting measurements of
(x t h in terms of tr .
Table 10 is a collection of thermal neutron (n,y) capture cross sections.
cr(jl,y) =
V
•rofof v 2 e" y '" ? dv
•Jo
^
(fth
r^'
 (TO
I v 3 e v'W dv
2
Sec. 2C]
Interaction with Nuclei
327
10,000
7
4
1000
<= 100
Is 7
1
Cd
/
/
\
Symbol en Reference No.
Figure in Legend
• 1
o 2
\
\
1
\
\
\
V
\
\
o
0.01
0.1
1.0
E n (ev)
(1) Rainwater, Havens, Wu, and Dunning, Phys. Rev., 71, 65 (1947). (2) C. T. Hibdon and C. O.
Muehlhause, Phys. Rev., 76, 100 (1949). Also see W. H. Zinn, Phys. Rev., 71, 575 (1947); Sawyer,
Wollan, Bernstein, and Peterson, Phys. Rev., 72, 109 (1947).
(a)
2
~ 1
E ioo
oo**.
I
n —
I
Hg
P V
°^oo
'°*a
1
2
♦ 3
■\o
N
^
•v,
^
V,
'<*■
»
VI
• . •
.
2
4
2
4
I
4
2
4
4
0.01
1.0
100
•BnCev)
1000
(1) W. W. Havens, Jr., and L. J. Rainwater, Phys. Rev., 70, 154 (1946). (2) L. B. Borst el al. (unpub
lished). (3) C. T. Hibdon and C. O. Muehlhause, Phys. Rev., 76, 100 (1949).
(b)
Fig. 20. Slowneutron cross sections showing the effects of a resonance at or near
thermal energies; from Adair (A2). (a) Cadmium, E r = 0.176 ev. Note the sharp
cutoff at ~0.30.5 ev. (b) Mercury, E r = —2.0 ev, showing the influence of a
close negativeenergy resonance.
328
The Neutron
[Pt. VII
TABLE 10
Radiative Capture Cboss Sections fob Slow Neutrons
Element
At
<r(n,y) t
(barns)
Neutron
Energy §
E r for Closest 
Observed
Resonance (ev)
References 1[
iH
1
0.313 ±0.013
»0
R20
2
0.00032
P
R20
3L1
7
0.033
th
W10, H68
4 Be
9
0.0085
to
620,000
R20
6 B
11
<0.05
th
430,000
W10
6 C
0.0045
th
~4, 000, 000
R20
13
~0.1
th
W10
14
<200
th
W10
7 N
15
0.000024
th
tt
8
0.00028 ±0.00022
P
440,000
R20
18
0.00022
th
W10, S22
9F
19
0.0094 ±0.0015
P
32,000
R20, S24
loNe
<2.8
P
tt
u Na
23
0.50
th,P
3,000
R20, H42, P23, C21
12M2
0.057
P
230,000
W10, H42, P23, C21
24
0.033
P
tt
25
0.27
P
tt
26
0.05
ih,P
S24, XX
isAl
27
0.22
P
2,300
W10, H42, P23, C21
l«Si
0.13
P.th
600,000
W10, H42, P23, T14, C21
28
0.08
P
»
29
0.27
P
tt
30
0.12
th.P
S24, tt
15P
31
0.17
P
W10, P23, C21
ieS
0.49 ±0.05
»o
111,000
R20
34
0.26
th
W10, S24
36
0.14
th
W10
17CI
32
P
W10, H42, P23, C21
35
42
th
75
H59
37
0.6
th
1,800
W10, S24
1»A
0.77
«o
>1,000
R20
40
1.2
P
W10
isK
2.0
P
65,000
W10, H42, C21, P23
39
~3
th
W10, H31
41
1.0
th
W10, S24
Sec. 2C]
Interaction wtih Nuclei
329
TABLE 10 (Continued)
Radiative Capture Cross Sections for Slow Neutrons
Element
At
(barns)
Neutron
Energy §
E T for Closest [
Observed
Resonance (ev)
References H
2oCa
0.41
P
~200,000
W10, H42, P23, C21
a
0.6
th
W10, S24
48
1
P
tt
2iSc
45
12
P
W10, P23
(20 s)
10
P
WW, G17
(85 d)
12
P,th
W10, S24, ft
22T1
5.6
P
W10, H42, C21, P23
46
0.57
P
tt
47
1.62
P
tt
48
7.98
P
tt
49
1.80
P
tt
50
0.14
P.th
W10, S24, }t
23V
4.7
P
WW, H42, C21, P23
51
4.5
th
2,700
W10, S24
24O
2.9
P
4,200
W10, R20, H42, C21, P23
50
16.3
P
W10, tt
52
0.73
P
W10, tt
53
17.5
P
W10, tt
54
0.006
P,th
WW, S24
2BMn
55
12.6
P
345
W10, H42, C21, P23
26^6
2.43
P
~10,000
W10, H42, C21, P23
54
2.1
P
P22
56
3.1
P
P22
57
0.5
P
P22
58
0.36
th
W10, S24
27CO
59
35
P
120
W10, H42, C21, P23
(10.7 m)
0.66
A
W10, S24
(5.3 y)
22
th
W10, S24
28 Ni
4.5
P
3,600
W10, H42, C21, P23
58
4.23
P
P22
60
2.70
P
P22
61
1.8
P
W10, tt
62
15
P
W10, tt
64
~2
ih
W10, S24
29C11
3.6
P
~500
W10, H42, C21, P23
63
4.29
P
W10, tt
65
2.11
P
W10, tt
30Z11
1.06
P
480
W10, H42, C21, P23
64
0.5
th
W10, S24
68
(13.8h)
0.9
a
W10, H72
(52 m;
0.1
th
W10, H72
70
0.085
h
W10, H72
330
The Neutron
[Pt. VII
TABLE 10 {Continued)
Radiative Captube Cross Sections fob Slow Neutbons
Element
At
(barns)
Neutron
Energy §
E r for Closest 
Observed
Resonance Cev)
References U
siGa
82Ge
S3A8
S4Se
SsBr
seKr
3?Eb
! 8 Sr
39Y
71
70
72
73
74
76
(59 s)
(12 h)
75
74
76
77
80 (17 m)
(59 m)
82 (67 s)
(25 m)
79 (18 m)
(4.4h)
81
78
80
82
83
84(4.4h)
(10 y)
90*
2.9
1.4
3.4
2.4
3.3
0.94
13.7
0.60
0.35
0.03
0.2
4.2
12.0
44
82
40
0.5
0.03
0.05
0.004
6.5
8.5
2.9
2.3
28
0.3
95
45
205
0.1
0.06
0.06
<470
0.70
0.7
0.12
<200
1.2
1.3
0.005
<110
~1
1.4
P
th
th
P
P
P
P
P,th
P
P,th
P,th
P
P
P
P
th
th
th
th
P
th
th
ih
P
th
P
P
P
th
th
th
th
P
th
th
th
P
th
th
P
P
100500
W10, H42, P23
W10, H72
W10, S24
W10, H42, P23
W10, tt
»
U
tt, S24
tt
A33, tt
S24, ft
W10, H42, P23, C21, ft
W10, H42, C21, P23
tt, S24
tt, A33
tt
W10, S24
W10, S24
W10, A33
W10, A33
W10, H42, P23, C21
W10, H72, tt
W10, H72, ft
W10, S24
tt
W10, tt
tt
tt
tt
W10, tt
W10, tt
W10, tt
W10, tt
W10, P23
W10, S24
W10, S24
W10
W10, H42, P23, C21, tt
W10, S24
W10, S24
W10
W10
W10, P23
Sec. 2C]
Interaction with Nuclei
331
TABLE 10 (Continued)
Radiative Capture Cross Sections for Slow Neutrons
Element
At
<K»,r) t
(barns)
Neutron
Energy §
E r for Closest I
Observed
Resonance (ev)
References H
«Zr
0.20
P.th
WW, R20, H42, P23, C21
90
~0.1
P
W10, tt
91
1.52
P
WW, it
93
~0.25
P
W10, n
94
~0.08
P
WW, it
96
~0.1
P
WW, tt
4lNb
93
1.1
P
WW, H42, C21, P23, tt
42M0
2.4
P
WW, H42, P23, C21, ft
92
<0.001
th
WW, S24
95
13.4
P
WW, tt
96
1.2
P
WW, it
97
2.1
P
WW, tt
98
~0.38
P.th
WW, S24, tt
100
~0.5
P.th
WW, S24, tt
44R11
2.5
P
9.4
WW, H42, P23, ft
96
0.01
th
W10, tt
102
1.2
th
WW, S24
104
0.7
th
WW, S24
«Rh
103
150
P
1.3
WW, H42, P23
(44 a)
137
th
WW, S24
(4.3 m)
12
th
WW, S24
46Pd
8
P
WW, H42, P23, C21
108
11
th
24
WW, S24
110
0.4
th
WW, S24
«Ag
60
P,th
R20, H42, P23, tt
107
30
P
45
P22
109
84
P
5.1
P22
(22 8)
~100
th
WW, S24, tt
(225 d)
2.3
th
WW, S24, tt
48Cd
2,400
»o
R20, tt
3,500
P
P23
106
1
th
tt
110
0.2
th
WW, G17
113
19,500
•0
0.18
WW, M39, D22, tt
114 (2.3 d)
1.1
th
WW, S23
(43 d)
0.14
th
WW, S23
116
1.4
th
~100
WW, S23
49ln
190
H,P
R20, P23
113 (72 s)
2.0
th
3.8
WW, G17
(50 d)
56
th
WW, S24
115 (13 s)
52
th
1.44
WW, S24
(54 m)
145
th
WW, 824
332
The Neutron
[Pt. VII
TABLE 10 (.Continued)
Radiative Capture Cboss Sections for Slow Neutrons
"(n,y) t
(barns)
Neutron
E r for Closest 
Element
a\
Energy §
Observed
References If
Resonance (ev)
soSn
0.65
P, »o
H42, P23, C21, R20
112
1.1
A
W10, S24
118 (279 d)
0.01
ft
W10, M29, B55
120
0.22
ft
W10, S24
122 (40 m)
0.30
ft
W10, S24
124 (10 m)
0.6
ft
W10, S24
(10 d)
0.15
ft
W10, S24
siSb
6.4
P
5.8
WIO, H42, P23, C21
121
6.8
ft
WIO, S24
123 (1.3 m)
0.03
ft
WIO, D13, tt
(21m)
0.03
ft
tt
(60 d)
2.5
ft
W10, S24
62Te
4.5
P
>300
W10, H42, P23, tt
120
68
P
tt
122
2.7
P
W10, H62, tt
123
390
P
tt
124
6.5
P
W10, H62, tt
125
1.5
P
tt
126 (9. 3 h)
0.8
ft
W10, S24
(90 d)
0.07
ft
W10, S24
128 (72 m)
0.13
ft
W10, S24
(32 d)
0.015
ft
W10, S24
130 (25 m)
0.22
ft
W10, S24
(30 h)
<0.008
ft
W10, S24
53I
127
6.7
P
20
WIO, H42, P23, C21
129*
~10
ft
W10, tt
131*
~600
ft
W10, tt
54X6
31
ft
R20
128
<5
P
tt
129
~45
P
tt
130
<5
P
tt
131
120
P
tt
132
0.2
ft
W10, tt
134
0.2
ft
W10, tt
135*
3.5 X 10 6
P
U2
136
0.15
ft
W10, tt
b&Ca
133
29
P
W10, H42, P23, R20, tt
(3h)
0.016
th
W10, S24
(2.3 y)
26
ft
WIO, S24
135*
~15
ft
W10, S57
137*
<2
ft
tt
6eBa
1.2
P
WIO, H42, C21, P23
130
~3
ft
WIO, K3, tt
132 (>20 y)
~6
ft
WIO, K3, tt
138
0.5
ft
WIO, S24
139*
4
Ik
W10, Yl, ft
Sec. 2C]
Interaction with Nuclei
333
TABLE 10 {Continued)
Radiative Captuke Cross Sections for Slow Neutrons
Element
A\
<r(.n,y) t
(barns)
Neutron
Energy §
E r for Closest 
Observed
Resonance (ev)
References U
srLa
8.9
P
W10, H42, P23
139
8.4
th
WIO, S24
140 «
~3
th
W10, K3
csCe
0.8
P
WIO, H42, P23
138
~0.4
th
tt
140
0.27
th
W10, K3, tt
142
0.85
th
W10, K3, tt
69Pr
141
11.2
P
(~10?)
W10, P23
eoNd
44
P
W10, H42, P23, tt
142
<12
P
W10, H55
143
240
P
W10, H55
144
<15
P
W10, H55
145
<30
P
W10. H55
146
1.8
th
W10, B54, tt
148
3.7
th
W10, B54, ft
150
<45
P
W10, H55
eiPm
147*
~60
th
W10, P4
B2Sm
6,500
to
R20
10,000
P
W10, H42, P23
149
~50,000
P
0.096
W10, tt
151
~7,000
P
tt
152
150
th
10
W10, S24, B54, tt
154
5.5
th
W10, S24
68Eu
4,200
P
W10, P23
151
~9,000
P
0.011
W10, H50, tt
(9 h)
1,400
th
W10, S24
152*
5,500
P
W10, H50, tt
153
420
P
0.465
W10, H50, tt
154*
1,500
P
WIO, H50, tt
155*
14,000
P
W10, H50, tt
64Gd
36,000
P
0.03
WIO, H42, C21, P23
152
<125
a
WIO, S24
155
70,000
p
WIO, L4
157
120,000
p
WIO, L4, tt
158
~4
th
WIO, B77, tt
180
~0.15
th
WIO, B77, tt
« S Tb
159
44
P
WIO, P23
eeDy
890
P
1.01;1.74
P23
1,150
»o
WIO, B65
164 (1.3 m)
3,000
th
WIO, tt
(2.4h)
2,600
th
WIO, S24, tt
165*
5,000
th
WIO, K10
TABLE 10 (Continued)
Radiative Captuee Ceoss Sections foe Slow Neutrons
Element
A\
<r(n,y) t
(barns)
Neutron
Energy §
E r for Closest 
Observed
Resonance (ev)
References IF
!7Ho
165
64
P
WlOy P23
esEr
166
P
~0.5
W10, P23
170
>7
th
W10, B54
6»Tm
169
118
P
W10, P23
voYb
36
P
W10, P23
168
30,000
th
W10, A35, tt
174
60
th
W10, A35, tt
176
6.5
th
W10, A35, tt
7lLu
108
P
W10, P23
175 (3.7 h)
25
th
W10, S24, tt
176
4,000
th
W10, S24, ft
72Hf
120
P
~1.0
W10, H42, P23, C21
177
500
P
tt
180
10
th
W10, S24
7sTa
181
21
P
4.1
W10, H42, P23, C21
(16 m)
0.030
th
W10, S24, tt
(117 d)
21
th
W10, S24
7»W
19
P
W10, H42, P23, C21
180
~2
th
L19
182
4
183
7.4
184
2.1
th
W10, S24
186
40
th
19.5
W10, S24, tt
187*
~80
th
L19
76Re
84
P
2.3
W10, P23, tt
185
100
th
tt
187
75
th
tt
760s
14.7
P
6.5
W10, P23
184
~20
th
L19
190
8
th
W10, S24
192
1.6
th
W10, S24
193*
~190
th
L19
77lr
440
P
0.64
W10, H42, P23
191 (1.5 m)
260
th
W10, G15, tt
(70 d)
740
th
W10, S24, tt
193
130
th
(1.3)
W10, S24
7 8 Pt
8.1
P
11.5
W10, H42, P23, C21, tt
192
90
th
W10, S24, tt
196 (18 h)
1.1
th
W10, S24
(82 d)
0.055
th
tt
198
3.9
th
W10, S24
334
TABLE 10 (Continued)
Radiative Capture Cross Sections for Slow Nbutbons
Element
At
»(n,7) t
(barns)
Neutron
Energy §
E, for Closest 1
Observed
Resonance (ev)
References *[
79A11
197
95
P.th
4.8
W10, H42, P23, S24, tt
198*
~16,000
th
ft
soHg
400
«o
G20
340
P
2.0
W10, H42, P23, tt
196
3,100
P
W10, 11
199
2,500
P
W10, 11
200
<60
P
W10, 11
201
<80
P
W10, 11
202
3.0
th
W10, S24, tt
204
0.43
th
W10, S24, tt
siTl
3.3
P
260
W10, H42, P23
203
8
th
W10, S24, tt
205
0.10
th
W10, S24, tt
82Pb
206
0.17
P
130,000
W10, H42, P23, C21, ft
208
0.0006
th
350,000
W10, tt
83B1
209
0.032
P
W10, H42, tt
(5d)
0.017
th
W10, S24, tt
joTh
232*
7.0
P.th
R20, tt
i 2 U
3.5
TO
TJ2
235*
101
no
D2
238*
2.80
m
11
U2
94PU
239 *
361
n
V2
96Am
241*
890
P
H29
(16 h)
570
P
H29
t The atomic number A refers to the target nucleus. When more than one activity results from the radiative capture
the halflife of the particular activity, to which the figures in that row pertain, is shown in parentheses. An asterisk
indicates that the target nucleus is itself radioactive.
% Unless otherwise indicated, the uncertainty can be assumed to be in the last significant figure.
§ The symbols have the following meanings:
to = neutrons of velocity 2200 m/sec (E = £§ ev)
P = pile neutrons, usually indicating measurement by the "danger coefficient" method
th = thermal neutron distribution for a temperature of ~300°K (usually indicates measurement by the "activation"
technique)
lj The closest observed resonance is not necessarily the closest resonance, since most elements have not been carefully
investigated above ~I0 ev. Negative resonances are shown when known.
H This compilation leans heavily on the excellent and complete collection of nuclear data by Way and coworkers
(W10). The compilation of Ross and Story (R20), although unfortunately somewhat out of date, has been most useful.
ft This table has been checked against a preliminary version of the extensive collection of the A.E.C. Cross Sections
Committee, D. J. Hughes, Chairman. A number of values from that table have been added to this one. Furthermore,
we have inclined somewhat to the choices, between alternative values, made by that committee. The responsibility
for the choices, however, rests on our shoulders. We gratefully acknowledge our indebtedness to D. J. Hughes and
his committee.
Jt We are grateful to H. Pomerance of the Oak Ridge National Laboratory for making available unpublished measure
ments of his group based on the "danger coefficient" method (P23)
335
336
The Neutron
[Pt. VII
(b) Intermediate Neutrons. For light and medium nuclei, (n,y) cross
sections for intermediate neutrons are very small, since T y « r„. In
heavy nuclei, the radiative capture cross sections are still quite appre
ciable in this energy range. The average (over many resonances) cross
section is given by Eqs. (60), (62), (62a), and (63) :
_ 2w^T n V y /600\ /iy\
500\/iy
II — I barns
(60a)
(E is in ev). Thus, the cross section follows a 1/v law at low energies,
and a 1/E law at high energies, the transition occurring when T n « T y ;
for heavy nuclei, this occurs in the intermediate region.
i
8
I
«
6
5
.(T t J
•
4
d
3
t
»1.0
H
St 8
®
■" r
, b
f
1
4
a
ol on Reference
No."
d
3
Sj
111U
Figure in Leger
• 1
2
x l l
© 4
o 5
8
6
5
4
3
0.1
^7„(Mev) 
10
(1) R. Fields e( at., Phys. Rev., 71, 508 (1947). (2) H. Aoki, Proc. Phys.Math. Soc. Japan, 21, 232
(1939). (3) J. R. Dunning et al., Phys. Rev., 48, 265 (1935). (4) E. Segre et erf. (unpublished). (5)
J. Marshall and L. Szilard (unpublished).
Fig. 21. Radiative capture cross section, a(n,y), for iodine in the intermediate
neutron energy range. Also shown is the total cross section, 07. From Goldsmith,
Ibser, and Feld (G20).
(n,y) cross sections for intermediate neutrons have not been so exten
sively investigated as those for slow neutrons. Among the light and
medium elements, fluorine, aluminum, and vanadium have been studied
by Barschall and coworkers (see the compilation of Adair, A2); the
<r(n,y) vs. E curves show the expected resonances; the magnitude of
Sec. 2C] Interaction with Nuclei 337
<j(n,y) is small — of the order of millibarns. Curves of a{n,y) vs. E have
been obtained, for a number of heavy elements, by Segre and coworkers;
their results are included in the compilation of Goldsmith, Ibser, and
Feld (G20). One of these, for iodine, is reproduced in Fig. 21. The cross
section shows the expected behavior (see Eq. 60a). Measurements on
fifteen isotopes, using SbBe neutrons (E ~ 35 kev), have been reported
by Hummel and Hamermesh (H77).
(c) Fast Neutrons. In the fastneutron energy region, the behavior of
<r(n,y) will differ from that described in the previous section for two
main reasons: (1) With the inception of the possibility of inelastic scat
tering, the competition in the deexcitation of the compound nucleus
becomes even less favorable to the radiative process. (2) For fast neu
trons the possibility of compound nucleus formation by neutrons of
I > becomes appreciable, so that relationships such as Eq. (62b),
based upon I = capture only, are no longer strictly valid. This factor
tends to compensate for the decrease of radiative capture due to inelas
tic scattering, since the inelastic reemission of I > neutrons is ener
getically unfavored.
Hughes, Spatz, and Goldstein (H72) have made a systematic survey,
covering 32 isotopes, of (n,y) cross sections at an energy of ~1 Mev.
Their results are summarized in Fig. 22. The most significant features
of these results are: (1) a rapid increase of c{n,y) with A (roughly expo
nential), from «1 millibarn at A « 35 to =200 mb at A « 110; (2)
roughly constant (n,y) cross sections, of ~100 mb, for A > 120; (3)
marked deviations from the norm for target nuclei containing neutron
closed shells (e.g., Ba 138 , 82 neutrons; Pb 208 and Bi 209 , 126 neutrons).
These magic number nuclei have anomalously small (n,y) cross sections
of ~23 mb. Since the radiative capture cross sections at ~1 Mev are
essentially inversely proportional to the compound nucleus level spacing
at the excitation energy (e + 1) Mev (see Eqs. 60a and 37, with T
~ T n » T 7 ), the general dependence on atomic number is in reasonable
agreement with expectation. The anomalous behavior of the magic
number nuclei reflects their small binding energies for an additional
neutron and, possibly, large level spacing.
Hughes et al. (H72) have also surveyed the available data on the de
pendence of a(n,y) upon neutron energy, for energies between ~0.1 and
~10 Mev. While the results of the different investigators are in rather
poor agreement with regard to the absolute values of a(n,y), the relative
values seem to follow, roughly, a ~l/i? law. There are, however, a
number of unexplained exceptions, notably In 115 , in which o(n,y) ap
pears to be constant over the energy range 0.1 to 1 Mev.
338
The Neutron
[Pt. VII
There are almost no data available on radiative capture in the very
fast and ultrafastneutron regions. At these energies, a{n,y) is expected
to be very small for all nuclei.
20 40
100 120 140 160 180 200 220
1.0 
0.1
1 1 ' 1
i i
1 ■■T
— 1 1 1
OLu
o
OAg
n
S &
™ O An

Kl. .
oi
.. ° f
/^Ag
°Sb
, H B
/ #
•
^ J «
Br / ,
ONci
O / Nb
/ •*
•
V OK u
ONd
G^f.*^
*
OBf •
Gu/
Co°°/
1 Mo
9Kr°M<,
OPr
/S Cu
/ °Ni
OR u
LaS^
k>Mn
*
.■°Ce
OBi
okA
# /
KrA
OSr
o
OBa
.
/•$>v
OFb
Kb
~ /OA
OXe
. /ttci
MgO/
PA\
i i i
1 1
1 1
, 1 ,
40 60
100 120 140 160
A
180 200 220
Fig. 22. Activation (n,y) cross sections for fission neutrons (~1 Mev, average
energy) vs. atomic weight, A. The points lying appreciably below the smooth curve
all correspond to neutron numbers near or at one of the "magic" values — 50, 82, 126.
We are indebted to Hughes, Garth, and Eggler (private communication) for this
figure. Earlier results are reported and discussed by Hughes, Spatz, and Gold
stein (H72).
3. Charged Particle Reactions. After the capture of a neutron, the
compound nucleus can sometimes decay by emission of a charged parti
cle. Among the possible reactions, (n,p) and (n,a) are most frequently
encountered. The energy dependence of a charged particle reaction is
governed by Eq. (42) : a(n,a) = <r c r o /r.
(a) Slow Neutrons. For a charged particle reaction to take place with
slow neutrons, it is necessary that the reaction be exoergic (Q > 0).
Furthermore, if the reaction is to compete favorably, the available
energy must be sufficiently great to allow appreciable penetration of
the Coulomb barrier (the factor G a in Eqs. 32 and 37). These considera
tions limit the observable slowneutron (n,p) and (n,a) reactions to light
nuclei. The properties of the most important slowneutron charged
Sec. 2C]
Interaction with Nuclei
339
particle reactions are summarized in Table 11. The cross sections in
the third column refer to the isotopes involved in the reaction; these
must be multiplied by the relative abundances (fifth column) to obtain
the cross sections of the normal elements.
TABLE 11
Properties of Exoergic (n,p) and (n,a) Reactions
Reaction
Q Value
(Mev)
Isotopic Cross
Section at
v = 2.2 X 10 B
cm/sec (barns)
References
Relative
Abundance of
Isotope in
Normal
Element (%)
He 3 (»,p)H 3
Li 6 (n,a)H 3
B w (n,a)Li 7
N 14 (n,p)C 14
Cl 35 (n,p)S 36
0.7637
4.785
2.791
0.626
0.62
5060 ± 200
910 ± 100
3770 ± 110 t
1.76 ±0.05
~0.3
T16, C26, K14
T16, R20
T16, R20
T16, C26
W10
1  10 X 10" 5
7.4
18.83
100
75.4
f A more recent value of the B 10 (n,a) cross section is 3990 b (AEC Neutron
Cross Section Advisory Group, AECU2040, U. S. Department of Commerce,
May 15, 1952). This cross section is of special significance since a majority of
the quoted thermal neutron absorption cross sections, in this and in Table 10,
are based on a comparison with boron absorption. Thus a change in the ac
cepted value of this cross section is directly reflected in a change, of equal frac
tional magnitude, in many of the other values quoted.
The cross sections are given at a single neutron energy, 0.025 ev.
Since these reactions fully satisfy the conditions for Eqs. (58) and (59),
they can be assumed to follow the 1/v law in the slowneutron range.
Owing to the large level spacings of such light nuclei, the first resonances
occur well into the intermediate or even the fastneutron region. Fur
thermore, owing to the large Q values (available charged particle energy)
of these reactions, the reaction widths are essentially constant over a
wide energy range. Thus, these reactions obey the 1/v law over a com
paratively broad energy region which, for the B 10 (n,a) reaction, for in
stance, extends to > 10 4 ev.
(b) Intermediate Neutrons. For intermediate neutrons, charged parti
cle reaction cross sections depart from the 1/v law because (1) resonances
are present in or close to the intermediateenergy region, and (2) the
particle width, r , is no longer independent of the neutron energy. A
resonance in the cross section for the formation of the compound nucleus
is, of course, also a resonance in the reactions involved in the compound
340
The Neutron
[Pt. VII
nucleus decay, including the charged particle reactions (see, for in
stance, the compilation of Adair, A2, figures 8, 9, and 12).
An interesting example of charged particle reaction resonances is N 14 ,
which has been investigated with good resolution for neutron energies
between ~0.2 and 2 Mev, and is shown in Fig. 23. In addition to the
(n,p) reaction, previously discussed, the N 14 (n,a)B u reaction is also in
evidence. This reaction is slightly endoergic (Q= —0.26 Mev), and
does not have an appreciable cross section below ~1 Mev.
One of the most striking features of Fig. 23 is the apparent separation
of the resonances (corresponding to the decay of the same compound nu
cleus) into predominantly (n,p) — e.g., 1.4 Mev— and (n,a) — 1.8 Mev —
resonances. Although at first glance this may appear to be in contra
diction to the ideas of the compound nucleus picture — upon which we
have leaned so heavily — the observations are, as will be seen from the
following discussion, consistent with our present notions, if proper
account is taken of the angular momentum and parity properties of the
nuclear levels involved. 1
The nuclei involved are N 14 , C 14 , B 11 (all in their ground states), and
N 15 (in various excited states). The spins of the nuclei are: N 14 , 1=1;
C 14 , I = 0; B 11 , I = f; and their groundstate parities are: N 14 , prob
ably even (assumed 3 D, from the magnetic moment and to explain the
long halflife for the C 14 betadecay) ; C 14 , even; B 11 , probably odd (from
the magnetic moment). Assuming these parity assignments, the parity
of the level of N 1S , involved in the resonance, completely determines the
I value of the captured neutron, even I values being associated with the
states of even parity, and odd I values with odd states. From the laws
of conservation of parity and angular momentum, the lowest possible I
value of the emitted proton or alphaparticle is uniquely determined
according to the following scheme
involved) :
V 1 3
7 \ 2 2
(J is the spin of the N 15 state
7
2 .. .. l p
■i 1 • • * • la
1
1 1 3 .. l p
2 2 .. l a
2
2 2 4 l p
1 1 1 3 L
1 The author is indebted to Professor J. M. Blatt, who first called to his attention
this possibility for explaining the N 14 resonance separation. A similar discussion
has been presented by Johnson and Barschall (J5).
Sec. 2C]
Interaction with Nuclei
341
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342 The Neutron [Pt. VII
It is evident that for lowneutron energies and, correspondingly, low
energies of the emitted proton and (especially) alphaparticle (so that
small I values are favored for both incoming and outgoing particle) the
resonances divide into two groups: those for which J = \, favoring pro
ton emission; levels with J > § , which favor alphaemission.
The same sort of arguments can be carried through for different
assumptions concerning the parities of the nuclei. In particular, the
assumption of odd parity for the N 14 ground state (other states same
as above) leads to a reversal in the division between protonfavored and
alphafavored levels; i.e., the absorption of a thermal neutron {l n = 0)
is followed by the emission of an I = 1 proton, etc. Unfortunately, the
available data do not permit a choice between the two possibilities for
the parity of N 14 (J5, J7).
(c) Fast Neutrons. As the energy available to the charged particle
becomes greater, the Gamow barrier penetration factor approaches 1,
and charged particle emission is less inhibited. Thus, reactions which,
although exoergic or only slightly endoergic, have very small cross sec
tions for slow and intermediate neutrons become appreciable in the
fastneutron region. The N 14 (n,a) reaction, discussed above, is one
such case. Another example is Ne 20 (w,a)O 17 (J6).
There are a number of endoergic charged particle reactions whose
thresholds, E t = — Q(A + 1)/A, fall in the fastneutron energy range.
The cross sections for these reactions have a characteristic energy de
pendence, rising rapidly (from o = at the threshold) to a moreorless
constant value for energies greater than the "height" of the Gamow
barrier. Figure 24 shows the measured (»,p) cross sections of two reac
tions (on P 31 and S 32 ) whose thresholds fall at ~1 Mev.
The energy dependence of such "threshold reactions" is, at least for
energies below the barrier heights, primarily determined by the probabil
ity for penetration of the Coulomb barrier by the emerging charged
particle. The barrier penetration factor is a monotonically increasing
function of the available energy (E — E t ) and, for a given value of E,
strongly dependent on the angular momentum of the charged particle;
the smaller the angular momentum, the greater the penetration factor.
The energy dependence of the penetration factor has been discussed by
a number of authors (B24, B34, B40), and by Morrison in Part VI. In
general, if the reaction can proceed with the emission of charged parti
cles of zero angular momentum, it will prefer to do so, and the energy
dependence near threshold will be determined by Go, the Gamow factor
for I = particles. In this case the barrier height is
zZe 2
Bo = — (80)
Sec. 2C]
Interaction with Nuclei
343
where z and Z are the atomic numbers of the outgoing particle and
product nucleus, respectively, and the nuclear radius is given by (B34) :
and
R = 1.474* X 10~ 13 cm for protons
R = (1.304* + 1.2) X 1(T 13 cm for alphaparticles (80a)
Table 12 lists the properties of a number of useful (n,p) and (n,a) thresh
old reactions computed on the assumption of I = outgoing particles.
TABLE 12
Properties or FastNeutron Threshold Reactions
Based on computations by Feld, Scalettar, and Szilard, (F8), and Kiehn, (Kll)
Product
E (Mev)
E (Mev)
Reaction
Nucleus
E t (Mev)
for
for
E t + £o
HalfLife
Go = 0.1
Go = 0.5
(Mev)
P 31 (n,p)Si 31
2.7h
0.97
2.8
3.8
5.3
S 32 (n,p)P 32
14.3 d
0.96
3.0
4.1
5.6
AF(w,p)Mg 27
10 m
1.96
3.5
4.5
5.9
Si 28 (n,p)Al 28
2.3m
2.7
4.4
5.4
6.9
Fe 66 (w,p)Mn 56
2.6h
3.0
6.3
7.6
9.4
P 31 (»,a)Al 28
2.3m
0.91
6.6
8.3
9.8
AF(n,a)Na 24
14.9 h
2.44
7.5
9.1
10.9
The actual energy dependence of a given charged particle reaction is
determined, in addition to the barrier penetration factor (in T a ), by the
cross section for the formation of the compound nucleus, <r c . Among
the important aspects of the dependence of <r c on E, the presence of
resonances — especially in the light and medium nuclei, with which we
are concerned — will be reflected in the reaction cross section; some of
the resonances in a c may appear weakly or not at all in the reaction
cross section because of the angular momentum and parity properties
of the levels involved.
Although the Gamow penetration factor for the outgoing charged
particle becomes relatively constant (it slowly approaches one but is
prevented from being strictly constant or equal to one by the increasing
importance of higher angular momenta and their associated angular
momentum barriers) after the available energy exceeds the barrier
height, B , the reaction cross section will not remain constant as the
neutron energy is indefinitely increased. In the very fast and ultrafast
344
The Neutron
[Pt. VII
8
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ieference No.
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(1) R. Fields ef a*., Phys. Rev., 71, 508 (1947). (2) H. Aoki, Proc. Phys.Math. Soc. Japan, 21, 232
(1939). (3) E. D. Klema and A. O. Hanson, Phys. Rev., 73, 106 (1948). (4) R. F. Taschek (unpub
lished). (5) E. Bretscher et al. (unpublished).
Fig. 24. The (n,p) cross sections of P 31 and S 32 in the fastneutron region. Meas
urements between 1.4 and ~6 Mev, with poor energy resolution; these curves illus
trate the "threshold" behavior of endoergic charged particle reactions. Curves
from Goldsmith, Ibser, and Feld (G20).
1 1 a (n,p)
P SI (n,p)Si SI
1.5
> ^ a (n,p)
30l(T 26 cm 2 T
S S2 (n,p)P s
3.5 MeV
3.0 3.5 MeV
Fig. 24 (Continued). Good resolution measurements between ~2 and 3.5 Mev,
showing the effect of resonances in the compound nucleus. Curve for P 31 due to
Ricamo (R9); S 32 curve due to Liischer, Ricamo, Scherrer, and Zunti (L25).
345
346 The Neutron [Pt. VII
neutron energy regions additional reactions, involving multiple neu
tron as well as charged particle emission, become energetically possible,
and the cross section for any given charged particle reaction will, in
the ensuing competition, decrease with increasing neutron energy.
4. Fission. For nuclei of A > 130 the binding energy per nucleon de
creases gradually with increasing A (Section 2B1). Consequently, most
of the heaviest nuclei are energetically unstable against division into two
fragments, i.e., M(A) > M(A — a) + M(a), provided that one of the
fragments is a relatively tightly bound particle such as an alphaparticle
or O 16 nucleus. However, as is well known, spontaneous nuclear disinte
grations are only (with two exceptions) observed in the heaviest, the
naturally radioactive, elements. (Many nuclei, lighter than lead, which
disintegrate by alphaparticle emission, have been produced artificially;
but their lifetimes are much too short for them to be found in nature,
even if they had been present in the original distribution of the elements.)
It is, of course, the Gamow barrier against charged particle emission
which impedes the spontaneous disintegration of the heavy elements,
with the effect that alphaemission is a relatively longlived process and
the emission of heavier fragments proceeds at an unobservably slow rate.
There is, however, one mode of disintegration which involves the
release of such a large amount of energy that the disintegration can, for
the heaviest nuclei, proceed over the top of the barrier. This process is
fission, or the division of the nucleus into two approximately equal
fragments.
Consider a heavy nucleus, say Z = 92, A = '240. The binding energy
per nucleon is =7.5 Mev. Division into two fragments, e.g., Z t = Z 2
= 46 and A 1 = A 2 = 120, produces two nuclei which are approximately
at the peak of the binding energy curve with, however, a considerable
neutron excess, so that their average binding energy per nucleon is some
what less than that of the stable nuclei in this region, say = 8.3 Mev.
The energy release is then ~240 X 0.8 ~ 200 Mev. The barrier height
against separation of the fragments is
ZiZ%e Z\Z 2
R^+R 2 = A x v * + A 2 A
B = „ = ^ u ( 2mc2 ) ~ 20 ° M ev (80' )
[We have used the approximation R = (e 2 /2mc 2 )A H .] However, be
cause of the large charge of the fragments, the barrier is very "wide,"
and the lifetime against fission is a very strong function of the differ
ence between the barrier height and the available energy, being ex
ceedingly long for barriers only a few (~510) Mev higher than the
reaction energy, and exceedingly short for energies above the barrier.
Sec. 2C] Interaction with Nuclei 347
The greater the atomic number of the nucleus, the smaller is the margin
of barrier height over reaction energy. This increase of instability with
atomic number places an upper limit on the possible Z of the heaviest
nuclei which can be found among the naturally occurring elements;
nuclei with Z > 100 would be unstable against spontaneous fission
(B46). 1
For the heaviest of the naturally occurring nuclei, then, stability
against fission depends on the slight deficiency in the available energy
as compared to the barrier height. The addition of only a relatively
small energy can serve to push the reaction over the barrier. The ab
sorption of a neutron is one means of supplying energy, since capture is
accompanied by release of the neutron's binding energy. If the binding
energy is sufficiently great, as in the addition of a neutron to some odd
neutron nuclei, fission can be induced by the capture of a thermal neu
tron. Among the nuclei which undergo fission after thermal neutron
capture are IT 233 , U 235 , Pu 239 (S42), Am 241 (C35), and Am 242 (H30).
Since the energy release is so great, the fission widths, T f , for such
nuclei should be essentially independent of the neutron energy. Thus,
in the absence of close resonances, the thermal neutron fission cross
section should follow a 1/v law. However, for such heavy nuclei the
compound nucleus level spacing is expected to be relatively small. For
U 235 , a t >,(n ,fiss.) = 545 barns (U2); additional data are given in Sec
tion 4D2.
In most of the heaviest nuclei, however, the energy release accompany
ing neutron capture is insufficient to cause fission. For such nuclei,
fission can be induced only if the neutron also carries with it a certain
amount of kinetic energy, so that E + e + Q > B . The fission cross
sections of these nuclei exhibit comparatively sharp thresholds. Many
of the thresholds are in the fastneutron region, and the corresponding
nuclei make excellent "threshold detectors." Some of these nuclei, and
their fission thresholds, are collected; in Table 13. Curves of o fes . vs. E
for U 238 and Np 237 are shown in Fig. 25.
TABLE 13
Approximate Values of Fast Fission Thresholds
Threshold
Threshold
Nucleus
(Mev)
Nucleus
(Mev)
Bi 209
60
TJ238
1.1
Th 232
1.3
Np 237
0.4
Pa 232
0.5
1 The experimental data on spontaneous fission are summarized by Segre [Phys.
Rev., 86, 21 (1952)].
348
The Neutron
[Pt. VII
1.5
Np 23
7
•
" m
•
V
*f
0.5
0.5
1.0
1.5
E (Mev)
(a)
2.0
0.7
0.6
SO6
8 0.3
tj2S8
0.2 0.3 0.4 0.6 0.8 1.0 2.0
Neutron energy (Mev)
(b)
3.0 4.0 6.0
F'g. 25. Fast fission cross sections vs. neutron energy, illustrating the "threshold"
nature of the fission reaction, (a) Np 237 . From E. D. Klema, Phys. Rev., 72, 8S
(1947). (b) Natural uranium; threshold due to U 238 fission (U2).
Sec. 2C] Interaction with Nuclei 349
In the very fast and ultrafastneutron regions, fission can be induced
in nuclei of smaller atomic number. Thus, nuclei down to platinum
have been shown to undergo fission on bombardment by 84Mev neu
trons (K7).
The fission process has a number of unique features. In addition to
the relatively tremendous amount of energy released per fission (Q ~
200 Mev), the process is also accompanied by the emission of neutrons
since the fragments, which are comparatively neutronrich, are emitted
with sufficient excitation energy to evaporate one or more neutrons.
It is this feature which has made possible the achievement of a nuclear
"chain reaction."
Thermal neutron fission is actually asymmetrical, i.e., the two frag
ments have a tendency to be unequal in charge and weight, with the
peaks in the fragment distribution occurring in the regions of A ~ 100
and A ~ 140. This tendency toward asymmetry is probably associated
with the greater stability of nuclei containing the neutron magic num
bers 50 and 82 (G7). For fission induced by ultrafast neutrons, on the
other hand, the fragment distribution appears to be more nearly sym
metrical. The change from asymmetry to symmetry with increasing
bombarding energy is not inconsistent with an explanation in terms of
the stability of magic number nuclei: a very highly excited nucleus will,
before undergoing fission, evaporate a considerable number of neutrons;
it will, therefore, not contain the ~132 neutrons necessary to produce
nuclei close to two different magic numbers but will, rather, tend to
split in such a way as to leave both fragments with as close as possible
to 50 neutrons, i.e., symmetrically.
5. The (n,2ri) Reaction. The preceding discussion has covered all the
exoergic reactions, and a few of the endoergic possibilities (inelastic
scattering, charged particle emission), which can be induced by slow
and intermediate neutrons. As the neutron energy is increased, through
the fast, very fast, and ultrafast regions, a number of other threshold
reactions become possible. One of the most important of these is the
(n,2n) reaction, whose Q value is the binding energy of a neutron in the
target nucleus. The product nucleus is an isotope of the target; in many
cases it is radioactive, frequently decaying by positron emission.
Neutron binding energies vary throughout the periodic table from
1.67 Mev in Be 9 to ~20 Mev in C 12 . Table 14 is a compilation of
(n,2ri) thresholds. Most of the thresholds have been obtained from ob
servations on (y,n) reactions which, starting from the same target and
leading to the same product nucleus, have the same Q values as the
corresponding (n,2n) reactions.
350
The Neutron
[Pt. VII
TABLE 14
Thresholds fob (y,ri) and (n,2ri) Reactions; Neutron Binding Energies
Target
Element
A
(7,™)
Threshold t
(Mev)
(n,2re)
Threshold t
(Mev)
Product Nucleus
HalfLife t
References
iH
2
2.226 ±0.003
3.34
stable
M34
3
6.25 ±0.01
8.33
stable
K16
3L1
6
5.35 ±0.20
6.2
S26
7
7.15 ±0.07 §
8.2
stable
S26
4 Be
9
1.666 ±0.002
1.85
5 X 10
14 „
s a
M34
6 B
10
8.55 ±0.25
9.4
S26
11
11.50 ±0.25
12.6
stable
S26
eC
12
' 18.7 ±0.1 §
20.3
21m
3+
B6, M13
?N
14
10.65 ±0.2
10.54 ±0.1 §
11.3
10 m
P +
M13
01
sO
16
16.3 ±0.4
17.3
2m
f> +
B6
9 F
19
10.40 ±0.3 §
10.9
1.9h
H +
01
uNa
23
12.05 ±0.2
12.6
2.6y
+
S26
liMg
24
16.4 ±0.2
17.1
12 s
3 +
M13, S26
25
7.25 ±0.2
7.5
stable
S26
26
11.15 ±0.2
11.6
stable
S26
13AI
27
12.75 ±0.2
13.2
7s
fi +
S26, M13
i<Si
28
16.8 ±0.4
17.4
5s
P +
M13
29
8.45 ±0.2
8.7
stable
S26
15P
31
12.20 ±0.2
12.6
25 m
/?+
M13, S26
ibS
32
14.8 ±0.4
15.3
3.2s
+
M13
34
10.85 ±0.2
11.2
stable
S26
17CI
?
9.95 ±0.2
10.2
S26
19K
39
13.2 ±0.2
13\5
7.7 m
3 +
M13
2oCa
40
15.9 ±0.4
16.3
Is
M13
22T1
46
13.3 ±0.2
13.6
3.1h
f) +
02
48?
11.6 ±0.3
11.8
stable
S26
49
8.7 ±0.3
8.9
stable
S26
23V
51
11.15 ±0.2
11.4
stable
S26
uCv
50
13.4 ±0.2
13.7
42 m
B +
02
52
11.80 ±0.25
12.0
25 d
K
S26
53
7.75 ±0.2
7.9
stable
1
S26
Sec. 2C]
Interaction with Nuclei
351
TABLE 14 (Continued)
Thresholds for (y,n) and (n,2n) Reactions; Neutron Binding Energies
Target
Element
A
Threshold t
(Mev)
(n,2n)
Threshold t
(Mev)
Product Nucleus
HalfLife %
References
25MH
55
10.1 ±0.2
10.3
310 d K
H35, S26
26Fe
34
13.8 ±0.2
14.1
8.9 m /3 +
M13
56
11.15 ±0.25
11.3
2.9y
S26
57
7.75 ±0.2
7.9
stable
S26
27C0
59
10.25 ±0.2
10.5
72 d, 9.2 h K, + , IT
S26
2sNi
58
11.7 ±0.2
11.9
36 h 0+
01
61?
7.5 ±0.3
7.6
stable
S26
29CU
63
10.9 ±0.2
11.1
10 m /3 +
M13, S26
11.2
11.4 ±0.3
F40
65
10.0 ±0.2
10.2
12.8 h + , /3~, K
M13, S26
3oZn
64
11.7 ±0.2
11.9
30 m +
H35, S26
66
11.15 ±0.2
11.3
250 d K, j3 +
S26
67
7.00 ±0.2
7.1
stable
S26
68
10.15 ±0.2
10.3
stable
S26
70
9.20 ±0.2
9.3
52 m p~
H35
3iGa
69
10.10 ±0.2
10.2
68 m (S + , K
S26
71
9.05 ±0.2
9.2
S26
33AS
75
10.2 ±0.2
10.4
17 d + , /3~, K
01, S26
34Se
82
9.8 ±0.5
9.9
18 m 0~
B6
>9.8
57 m IT
B6
?
7.30 ±0.2
7.4
S26
?
9.35 ±0.2
9.5
S26
3sBr
79
10.65 ±0.2
10.9
6.3 m 3 +
M13, S26
81
10.1 ±0.2
10.3
18.5 m pT, P +
M13, S26
3sSr
86
9.50 ±0.2
9.6
65 d K
S26
87
8.40 ±0.2
8.5
stable
S26
88
11.15 ±0.2
11.2
stable
S26
4<)Zr
90
12.48 ±0.15
12.6
4.5m K01IT
H35
12.0 ±0.2
12.1
78 h ,8+
01
91
7.2 ±0.4
7.3
stable
H35
4iNb
93
8.70 ±0.2
8.8
10 d /3 
S26
42M0
92
13.28 ±0.15
13.4
16 m, 17 s (3 +
H35
97
7.1 ±0.3
7.2
stable
H35
?
6.75 ±0.25
6.8
S26
?
7.95 ±0.25
8.1
S26
44RU
?
7.05 ±0.2
7.1
S26
?
9.50 ±0.2
9.6
S26
352
The Neutron
[Pt. VII
TABLE 14 (Continued)
Thresholds for (y,n) and (w,2n) Reactions; Neutron Binding Energies
Target
Element
A
(y,n)
Threshold t
(Mev)'
(n,2n)
Threshold t
(Mev)
Product Nucleus
HalfLife J
References
4sRh
103
9.35 ±0.2
9.4
210 d pr, 0+
S26
4 6 Pd
?
7.05 ±0.2
7.1
S26
?
9.35 ±0.2
9.4
S26
47Ag
107
>9.5
24.5 m 0+
B6
109
9.05 ±0.2
9.1
2.3 m /3 
S26, B6
4«Cd
113
6.5 ±0.15
6.6
stable
H35, S26
4»In
115
9.05 ±0.2
9.1
50 d IT; 72 a /3 _ , K, 0+
S26
5oSn
118
9.10 ±0.2
9.2
stable
S26
119
6.55 ±0.15
6.6
stable
H35. S26
124
8.50 ±0.15
8.6
40 m /3~
H35
5lSb
121
9.25 ±0.2
9.3
17 m /3 +
M13
?
8.95 ±0.25
9.0
S26
123
~9.3
~9.4
2.8d /3~
J4
52 Te
?
6.50 ±0.2
6.6
S26
?
8.55 ±0.2
8.6
S26
5 si
127
9.3 ±0.15
9.4
13 d 3 
Ol, M13, S26
55Cs
133
9.05 ±0.2
9.1
7. Id K
S26
56Ba
J
6.80 ±0.2
6.8
S26
?
8.55 ±0.25
8.6
S26
57La
139
8.80 ±0.2
8.9
stable; 18 h K, IT
S26
ssCe
140
9.05 ±0.2
9.1
140 d K
S26
142
7.15 ±0.2
7.2
33 d /3~
S26
69 Pr
141
9.40 ±0.10
9.5
3.5 m /3 +
H35
eoNd
150
7.40 ±0.2
7.4
2h /3~
H35
7 3 Ta
181
7.6 ±0.2
7.6
8.2 h (T,K
M13, S26, J4
74W
?
6.25 ±0.3
6.3
S26
?
7.15 ±0.3
7.2
S26
75Re
187
7.3 ±0.3
7.3
93 h 0 K
S26
77lr
193
7.80 ±0.2
7.8
70 d 0
S26
7 S Pt
194
9.50 ±0.2
9.5
4d K
S26
195
6.1 ±0.1
6.1
stable
P5, S26
196
8.20 ±0.2
8.2
stable
S26
79AU
197
8.05 ±0.10
8.1
5.6d pr
P5, H35, S26
Sec. 2C]
Interaction with Nuclei
353
TABLE 14 (Continued)
Thresholds for (y,n) and (n,2n) Reactions; Neutron Binding Energies
Target
Element
A
(t,«)
Threshold t
(Mev)
(n,2n)
Threshold t
(Mev)
Product Nucleus
HalfLife J
References
soHg
201
6.25 ±0.2
6.6 ±0.2
6.3
6.6
stable
H35
P5
siTl
203
205
8.80 ±0.2
7.5 ±0.15
8.8
7.5
12 d K
3y 0
S26
H35, S26, P5
82Pb
206
207
208
8.25 ±0.10
6.88 ±0.10
7.40 ±0.10
8.3
6.9
7.4
stable
stable
PI, P6
PI, S26, P5
PI, S26
83B1
209
7.4 ±0.1
7.4
M13, P5, S26
9oTh
232
6.35 ±0.04
6.4
26 h p~
H76, P5
92U
238
5.97 ±0.10
6.0
6.8 d 0
H76, P5
t The (n,2n) thresholds are computed from the measured (y,n) thresholds (or vice versa) according
to the relationship Et(n,2n) = [(A + l)/AlEi(.y,n).
t The half lives and radioactivities are from the compilation of Way, Fano, Scott, and Thew (W10).
§ Thresholds computed from nuclear mass data and used to calibrate the gammaray energy scale.
The most recent data on the thresholds for light elements are summarized by F. Ajzenberg and T.
Lauritsen, Revs. Modern Phys., 24, 321 (1952).
In light nuclei, the (n,2n) reaction competes with charged particle
emission as well as with neutron scattering. In heavy nuclei, charged
particle reactions are so strongly inhibited by the Coulomb barrier that
the only important competing reaction — at least, for energies not too
far above the (n,2n) threshold—is scattering, mainly inelastic at these
energies. Assuming that this situation prevails, the value of the cross
section, <x{n,2n), can be computed, given the energy distribution of the
inelastically scattered neutrons, do(z,E),
T(n,2n)
e = EE t
d*(z,E)
(81)
That is, if the first neutron is emitted with sufficiently low energy
(s < E — E t ), the residual nucleus will still have enough excitation to
emit a second neutron and, since neutron emission is the most probable
mode of decay for heavy nuclei at high excitation energies, it will almost
always do so.
At the excitation energies involved, and especially for the lowenergy
part of the spectrum, the distribution of inelastically scattered neutrons
354
The Neutron
[Pt. VII
is given, to a good approximation, by the Maxwellian distribution
(Eq. 69). Thus,
fEE, / s \
a(n,2n) = a c J \j e *T de
(82)
= <r c [l  (1 + y)e~»]
where y = (E — E t )/T. For energies close to the threshold, i.e., y « 1,
r(n,2ri)
<fcy"
2T 2
{E  E t f
(82a)
The (E — E t ) 2 dependence of the (n,2n) cross section near threshold
has been verified by Fowler and Slye (F40) for the Cu 63 (n,2n) reaction.
1
1
.35
1
"
.30
~
.25
.20
(M

.15
s
fc5"

.10
.05
11.0
12.0
13.0
14.0
ll.U iz.u la.u 14.U
Neutron energy (Mey) (Uncertainty of energy scale ± 0.2_Mev)
Fig. 26. Cross section for the reaction Cu^n^n^Ou 62 in the vicinity of the threshold,
due to Fowler and Slye (F40). The lefthand scale gives the ratio of the Cu 63 (n,2n)
cross section to the Cu 66 (n,y) thermal neutron cross section. The absolute cross
section scale, on the right, is based on (0.56 ± 20%) barns for the thermal neutron
cross section of Cu 66 in normal copper; it should be increased by ~16 percent (pri
vate communication from Fowler).
Their crosssection measurements for neutron energies between E = E t
= (11.4 =fc 0.3) Mev and <~14 Mev are shown in Fig. 26. This reaction
is frequently used as a threshold detector of neutrons of energy E >
11.5 Mev.
The C 12 (w,2n) reaction has also been used as a threshold detector for
very fast neutrons. The cross section from threshold E t = 21 Mev to
~25 Mev increases rapidly with neutron energy in roughly the expected
manner (S27).
The expression for <x(n,2n), Eq. (82), has been derived on the assump
tion that the statistical theory, which leads to a Maxwell distribution of
Sec. 2C] Interaction with Nuclei 355
the emerging neutrons, is valid in the region of excitation under con
sideration. The cross sections near threshold bear out this assumption.
The statistical theory has been verified over a much wider range of
excitation energies by the measurements of Bradt and Tendam (B58)
on the relative cross sections for the (a,n) and (a,2n) reactions on silver
and rhodium with 1520 Mev alphaparticles, and by the observations
of Temmer (T7) and Kelly and Segre (K8) on the excitation functions
for the (a,ri), (a,2n), and (a,3n) reactions on indium and bismuth with
alphaparticles up to 40 Mev energy.
However, application of the statistical theory to (n,2n) reactions
induced by very fast (and, especially, ultrafast) neutrons is subject to
the limitation, pointed out by Weisskopf (W17), that neutrons of such
high energy have a large probability of penetrating through the nuclear
surface and hence may emerge from the nucleus before they have fully
shared their energy with the rest of the nuclear constituents. The energy
distribution of the emerging (first) neutron will, under these circum
stances, contain many more fast neutrons than predicted by the evapo
ration theory. This situation does not obtain for alphaparticle bom
bardment, since the Coulomb barrier impedes the emergence of the
alphaparticle and helps to achieve the sharing of energy in a true
compound nucleus.
In any event, if, after the emission of the first neutron, the residual
(target) nucleus is still sufficiently excited to emit a neutron, it will do
so. The energy distribution of the second neutrons is determined by the
level spacing of the product (final) nucleus at the excitation energy re
maining after emission of the second neutron. For incident neutron
energies not very far above the (n,2ri) threshold, the energy available to
the second neutron cannot be very large, and the spectrum of second
neutrons cannot be assumed to follow the Maxwell distribution. In
essence, the same considerations, which we have applied in the discus
sion of the energy distribution of inelastically scattered neutrons, will
apply in this case.
6. MultipleParticle Emission ; Spallation Reactions and Stars. The
interaction of very fast and ultrafast neutrons with nuclei can be con
sidered to take place in two relatively distinct stages:
(1) The initial interaction of the incident neutron with one or several
of the nuclear constituents. In this stage the nucleons can be treated as
relatively independent particles (S21), and the recoiling nucleons (as
well as the incident neutron) have a nonnegligible probability of leav
ing the nucleus, carrying off a relatively large fraction of the incident
energy.
356 The Neutron [Pt. VII
(2) Those nucleons which do not directly escape from the nucleus
will rapidly share their energy among the nuclear constituents. In gen
eral, the resulting excited nucleus will have sufficient energy to evapo
rate more than one particle.
The theory of the interaction of neutrons of E ~ 100 Mev with heavy
nuclei has been developed by Goldberger (Gil). He has computed the
energy and angular distributions of the emerging nucleons in the first
stage of the interaction, as well as the distribution of excitation energies
in the residual nuclei; his computations take into account the velocity
distribution of the nucleons in the target nucleus (which he treats as a
Fermi gas confined to the nuclear volume) .
The process of ejection of nuclear constituents through direct inter
action with the incident particle is sometimes referred to as spallation. 1
The second or evaporation part of the reaction is usually referred to as
star production, from the characteristic records that such reactions leave
in nuclear emulsions which are sensitive only to relatively lowenergy
charged particles.
One method of studying highenergy nuclear interactions is through
the yields of the various products, usually radioactive (K19). How
ever, aside from the obvious limitation of this method (that only radio
active products can be detected), it has the disadvantage that the two
stages of the reaction cannot be separated.
A more satisfactory method for studying such reactions is by observ
ing the emerging particles, either in a cloud chamber or in a sensitive
nuclear emulsion. In such investigations it is possible to separate the
two stages of the reaction by the difference in the energy and angular
distributions of the resulting charged particles. The products of the
first stage have high energy and forward collimation (Gil). Those of
the second stage have relatively low energy and are distributed uniformly
with respect to the direction of the incident particle (L12). The appli
cability of the Goldberger model to highenergy nuclear reactions has
been strikingly verified by the work of Bernardini, Booth, and Linden
baum (B19).
An interesting result of such investigations is the observation of a
considerable number of highenergy deuterons, emitted in the forward
direction (H2, B74, Y3, M37). These appear to result from a "pickup"
process in which the incident neutron is joined by a nuclear proton,
which happens to be moving in the same direction and with the proper
velocity, to form a deuteron (C13, H51, C34).
1 Spall : A chip or fragment, esp. of stone. Webster's Collegiate Dictionary, fifth
edition, G. and C. Merriam Co., 1948.
Sec. 3A] Sources and Detectors 357
The interaction of ultrafast neutrons with nuclei has been studied in
cosmicray investigations. Neutrons with energies exceeding 10 9 ev are
present in the cosmic radiation. Many of these investigations have
employed nuclear emulsions in which cosmicray stars are a common
phenomenon. Le Couteur (LI 2) has shown that the energy distribution
of the evaporation products in stars can be understood in terms of the
statistical model; the energy of the emitted protons and alphaparticles
is given by a modified Maxwell distribution, the modification arising
from the inhibition of emission of lowenergy charged particles by the
Gamow barrier.
Owing to the decrease with energy of the primary (n,p) and (n,n)
cross sections, nuclei are relatively transparent to ultrafast neutrons of
energy < 300 Mev. However, at neutron energies greater than the
threshold for meson production (~285 Mev for neutrons on protons or
neutrons at rest) the reaction cross section is expected to rise again.
The available evidence indicates that nuclei are essentially opaque (re
action cross section « wR 2 ) for neutrons of energy > 10 9 ev. The
products of the interaction of such neutrons with nuclei are, in the first
stage, a number (increasing with neutron energy) of mesons (probably
mostly pions) traveling in the forward direction, and fast nucleons
(E ~ 100 — 1000 Mev) also in the forward direction; the residual nu
cleus, in the second stage, usually has sufficient energy to evaporate a
large fraction of its particles (F10).
SECTION 3. SOURCES AND DETECTORS:
NEUTRON SPECTROSCOPY
A. Introduction
To investigate experimentally the properties of the neutron and its
various interactions, it is necessary to have a neutron source and a neu
tron detector. If, in addition, it is desired to confine the investigation to
neutrons of a specific energy, then either the source must emit mono
energetic neutrons or the detector must be sensitive only to neutrons of
a single energy. The problems of neutron spectroscopy, and the dis
cussions of this section, involve the choice of the proper sources or de
tectors or both in the various neutron energy ranges.
The resolution of a given experimental arrangement is a measure of
the energy spread of the neutrons selected by the source and detector.
As in other fields, good energy resolution must frequently be paid for
by decreased intensity, if it can be achieved at all. On the other hand,
poor resolution is by no means always an unmitigated ill. For example,
for a comparison of intermediate and fastneutron cross sections with
358 The Neutron [Pt. VII
the predictions of the continuum theory of Feshbach and Weisskopf, it
is necessary to average over many levels of the compound nucleus; such
an average is most easily and accurately achieved by using a source
whose energy spread is sufficient to cover many levels of the compound
nucleus.
The term "resolution" is usually employed in a descriptive, qualita
tive connotation, mostly in conjunction with an adjective, such as
"good" or "poor." The quantitative measure of energy resolution is the
resolution function, R(E) = S(E)D(E), the product of the source strength
and the detector efficiency. For many measurements, especially cross
section determinations in which there is considerable variation with
energy (e.g., resonances), a knowledge of R(E) is indispensable for the
interpretation of the experimental results. A considerable fraction of
the literature on neutron cross sections is essentially uninterpretable,
not so much because the experiments were performed with poor resolu
tion as because they were performed with sources of unknown energy
distribution or detectors with unknown efficiency curves.
However, although this criticism is sometimes applicable to recent
work, it is certainly not fair to apply it to most of the early investiga
tions. To the pioneers in neutron physics, in the preWorld War II
era, the number of sources and detectors available was severely limited.
Although many of their crosssection measurements may not have been
good in the presentday sense, or even quantitatively interpretable,
their exploratory investigations were invaluable, for only through the
accumulated knowledge of such explorations were presentday tech
niques made possible. In many instances what the early masters lacked
in technique was more than compensated by the ingenuity of their ex
periments and the penetration and insight of their interpretation.
The first available neutron sources were, naturally, those involved in
the discovery of the neutron: (a,n) reactions on many elements, espe
cially beryllium, using naturally radioactive alphaemitters. Some of this
early work has been described in the introductions to Sections 1 and 2.
The discovery of the photodisintegration of the deuteron by Chad
wick and Goldhaber (C8), and of beryllium by Szilard and Chalmers
(S63) uncovered another source of neutrons, the (y,ri) reaction on these
elements by gammarays from various natural radioactivities. These
sources differ from the (a,ri) sources in that they provide lowerenergy
neutrons, usually in the intermediateenergy range, since (y,ri) reac
tions are endoergic. On the other hand, such sources are usually con
siderably weaker than (a,n) sources. However, with the production of
strong artificially radioactive sources, at first through the use of parti
cle accelerators and more recently by neutron irradiation in nuclear re
Sec. 3A] Sources and Detectors 359
actors, a large variety of gammaray sources have become available for
photoneutron production. Furthermore, the development of high
energy electron accelerators for the production of intense xray beams
has added another means of intense photoneutron production, although
these sources are not monoergetic like those produced by a nuclear
gammaray.
The first extensive investigation of photoneutron production in
elements other than beryllium and deuterium was made by Bothe and
Gentner (B52). They utilized the <~17 Mev gammarays produced in
the Li 7 (p,y) reaction, using artificially accelerated protons of 0.4 Mev
energy. Since then, a large number of investigations of (y,n) reactions
have been carried out with these and other gammaray sources, and with
xray beams from electron accelerators. The results of many of these
are summarized in Table 14 (Section 2) and in the references therein
indicated. Of all the nuclei investigated, only beryllium and deuterium
have photoneutron thresholds of energy less than 6 Mev.
The development of highenergy heavyparticle accelerators led to the
discovery of a large number of new and useful neutronproducing reac
tions. In addition to providing monoenergetic alphaparticles of energy
higher than those available from the natural radioactivities, with which
it was possible to investigate further the (a,n) reactions on beryllium,
boron, and other elements, it was found possible to utilize other acceler
ated nuclei, such as the proton and the deuteron, for neutron production.
It was soon discovered that (d,ri) reactions have large yields at the
deuteron energies available from accelerators. Crane, Lauritsen, and
Soltan (C30) discovered the (d,n) reactions on lithium and beryllium.
Soon afterward, Lawrence and Livingston (L8) extended the investiga
tion of (d,n) reactions to a number of other elements. At about the same
time (all this in 1933 and 1934) Oliphant, Harteck and Rutherford (03)
discovered the d + d reaction; this reaction has been of great importance,
since it provides a strong source of monoenergetic neutrons with com
paratively lowenergy deuterons, enabling the use of relatively low
voltage particle accelerators for neutron sources. Other (d,ri) reactions
have comparable or higher yields at deuteron energies of a few Mev,
but the resulting neutrons are heteroenergetic (with the exception of
the d + t reaction).
Another reaction type of great importance to neutron spectroscopy is
the (p,n) reaction. Crane and Lauritsen (C31) discovered the ~Li(p,ri)
reaction, which has since been extensively used for a monoenergetic neu
tron source. However, this reaction does not really yield monoenergetic
neutrons, since it also gives rise to a second neutron group, due to an
excited state of Be 7 , 435 kev above the ground state. Fortunately, the
360 The Neutron [Pt. VII
second group has a comparatively low yield (<10 percent) for protons
up to ~4 Mev. A number of other (p,ri) reactions have been studied
and used from time to time for neutron sources, but none so extensively
as the Li 7 (p,n) reaction.
All the abovementioned neutron sources yield energies in the inter
mediate, fast, and very fast ranges. Until the advent of ultrahigh
energy accelerators, the only source of neutrons in the ultrafastenergy
range was from the cosmic radiation. The presence of highenergy
neutrons as a component of cosmic rays was established by the experi
ments of Rumbaugh and Locher (R25) and of Fiinfer (F51), and has
been used by many investigators to study the nuclear interactions in
duced by ultrafast neutrons. Although particle accelerators are at
present capable of producing neutrons of energy up to ~400 Mev, and
higherenergy accelerators (a few Bev) are now being constructed, cos
mic radiation still remains the only source in immediate prospect for
neutrons of energies of, say, > 10 Bev.
At the other extreme, the slowneutron region has been most exten
sively investigated. Sources of slow neutrons almost invariably have
their origin in fast neutrons, slowed down in paraffin or some other
material containing light elements. Since such sources yield a broad
distribution of neutron energies, it is necessary to employ some form of
neutron monochromator for studies requiring monoenergetic neutrons.
A number of monochromators have been, and are being, developed, and
their effective range has been slowly pushed up toward the intermediate
energy region, so that there now remains only a small gap between the
monoenergetic neutrons available from charged particle reactions and
from slowneutron monochromators. 1 The availability of very great
neutron intensities from nuclear reactors has provided a great impetus
to the development of more effective neutron monochromators of greater
range and flexibility.
The development of neutron detectors has rapidly followed the exten
sion of knowledge of neutron reactions; practically every new discovery
has led to a new means of neutron detection. Thus, the observation of
proton recoils by Curie and Joliot (C37), made even before the identifi
cation of the neutron as a new particle, led to the technique of observing
neutrons in ionization chambers, electroscopes, and cloud chambers by
lining these instruments with paraffin. The discovery of charged parti
cle reactions in lithium, boron, and nitrogen enabled the detection of
1 Actually, the gap is being closed from the intermediateenergy end as well. Thus,
Hibdon, Langsdorf, and Holland [Phys. Rev., 85, 595 (1952)] have succeeded in
studying the 225 kev range with an energy resolution of 2 kev, using the Li(p,w)
reaction.
Sec. 3B] Sources and Detectors 361
neutrons through the incorporation of these substances in ionization
chambers, proportional counters, and cloud chambers. These and other
reactions, as well as proton recoils, can also be observed in nuclear
emulsions.
The discovery of neutroninduced radioactivity provided still another
means of neutron detection which could be used to investigate specific
energies or energy ranges — thermal neutrons through 1/v capture cross
sections, specific slowneutron energies through various resonances,
fast neutrons by means of threshold reactions. As neutron reactions
have been further understood, and as various techniques of charged
particle counting have been improved and extended, the variety of neu
tron detectors has increased until now it is possible to find a suitable
neutron detector at almost any energy.
This is not to say that detector problems are negligible in neutron
studies; as is so often the case, ease of detection is in direct proportion to
the available intensity, so that the source and detector problems of
neutron spectroscopy go hand in hand. Especially in the fast, very
fast, and ultrafastneutron energy ranges, detectors are of relatively low
efficiency, and available sources are never quite strong enough, so that
neutron spectroscopy in these energy regions still presents difficult
problems.
Nevertheless, it seems fair to summarize by saying that the available
techniques of neutron spectroscopy allow an almost complete coverage
of the range of energies from to ~300 Mev with relatively few signifi
cant gaps, and that, with foreseeable extensions of available techniques,
the existing gaps should soon be closed.
B. Neutron Sources
1. Radioactive (a,ri) Sources. The discovery of the neutron involved
the reaction
4 Be 9 + 2 He 4 » 6 C 12 + on 1
induced by bombarding beryllium with aparticles emitted by the natu
rally radioactive elements. Although many other neutronproducing
reactions have since been discovered, the above reaction is still the basis
for some of the most extensively used neutron sources.
According to available mass values (T16, B2), this reaction is exoergic,
with a Q value of 5.65 Mev. Thus, starting with the polonium alpha
particles (energy 5.30 Mev), the emergent neutrons should have a spread
of energies between 10.8 Mev (outgoing neutron in the same direction
as the incoming alpha) and 6.7 Mev (outgoing neutron in the opposite
direction from the incoming alpha).
362 The Neutron [Pt. VII
However, the neutrons observed in the bombardment of beryllium
with polonium alphaparticles have a considerably greater energy
spread, ranging from the above maximum to energies well below 1 Mev.
The observed energy spread arises from two causes: (1) In the above
reaction, the C 12 nucleus may sometimes be left in an excited state, re
sulting in less available energy for the outgoing neutron. This possi
bility will, for monoenergetic incident alphaparticles, result in the
appearance of groups in the spectrum of the outgoing neutrons, each
group corresponding to an excited state of the C 12 nucleus. Evidence
for the existence and energy values of the C 12 levels is summarized by
Hornyak, Lauritsen, Morrison, and Fowler (H67). (2) In most (a,n)
sources, the thickness of the beryllium target is large compared to the
range of the impinging alphaparticles. (The range of a polonium
alphaparticle is 3.66 cm in standard air.) Since the cross section for
the neutronproducing reaction is small compared to the cross section
for energy loss by collisions with atomic electrons, very few nuclear
processes occur while the alphaparticle has its full, initial energy.
Thus, even if all the reactions led to the ground state of C 12 , the out
going neutrons in the forward direction would have an energy spread
ranging from the maximum (10.8 Mev) down to 5.2 Mev (correspond
ing to zero incident alphaparticle energy).
The energy spectrum of PoaBe neutrons is further complicated as a
result of the variation of the reaction cross section with the incident
alphaparticle energy. Thus, the necessity for the alphaparticle to
penetrate through the potential barrier of the beryllium nucleus de
creases the neutron yield for lowenergy alphaparticles; the height of
the potential barrier is ~3.7 Mev. Furthermore, the level structure of
the compound nucleus, C 13 , leads to resonances in the cross section
(H67).
As a result of the effects discussed above, the neutron spectrum from
a (thick target) PoaBe source is complex, and cannot be predicted in
detail. A number of attempts have been made (A27) to measure the
neutron spectrum from such a source using proton recoils, in nuclear
emulsions, as a neutron detector. The result of a recent measurement,
due to Whitmore and Baker (W20), is shown in Fig. 27.
Among the alphaemitting radioactive elements, polonium is compara
tively difficult to obtain in quantities sufficient to produce strong neu
tron sources. The elements radium and radon are, however, commer
cially available in sufficient quantity so that they are most frequently
used for neutron sources. Radium has the advantage of a very long half
life (~1600 years, as compared to 3.825 days for radon and 138 days for
polonium), which makes it particularly suitable for longlived sources.
Sec. 3B]
Sources and Detectors
363
On the other hand, polonium and, even more so, radon require a con
siderably smaller mass of beryllium in the source mixture to approxi
mate a thick target (since the mass per unit radioactive strength of a
radioactive element is proportional to its halflife). In particular,
radon requires quite small quantities of beryllium for high neutron
yields, and presents no difficulty in mixing, since it is a noble gas and
diffuses uniformly through powdered beryllium; thus, RnaBe sources
\
V
T N
4 5 6 7
Neutron Energy (Mev)
10
11
Fig. 27. Energy distribution of neutrons from a PoaBe source (W20).
can be made quite small. However, owing to the inconvenience of
working with a gas of such short halflife, radon is now seldom used for
neutron sources.
PoaBe sources have the added advantage that in the decay of
polonium there is a comparatively negligible gammaray emission,
which makes the handling of such sources relatively simple. The decay
products of radium, on the other hand, emit a prodigious quantity of
gammaradiation, and suitable precautions for protection against these
radiations must be observed in the handling of these sources. Never
theless, owing to the conveniences of availability and long life, RaaBe
neutron sources are very widely used.
Another disadvantage of polonium as compared to radium is that it is
more difficult to manipulate, despite the absence of gammaradiation.
One method of preparation of a PoaBe source, described by Spinks and
Graham (S49), consists in sandwiching a platinum foil, on which the
polonium is deposited, inside a cylinder of beryllium. Such a source
364 The Neutron [Pt. VII
has a smaller yield per curie (~3^) than an intimate mixture, and the
neutron emission is not isotropic.
During World War II considerable experience was obtained in the
preparation of radiumberyllium mixtures. 1 The preparation and
handling of such sources has been described by Anderson and Feld
(A26) and, in greater detail, by Anderson (A27). Most of these con
sist of an intimate, physical mixture of radium bromide and beryllium
metal powder, pressed into pellets of density ~1.75 g/cm 3 . In addition
to their small size (thereby more closely approximating a point source),
pressed sources are more likely to remain constant in time because of
the greater physical stability of the mixture.
The neutron spectrum from RaaBe sources is even more complex
than that of PoaBe, owing to the variety of alphaparticles emitted
by radium and its decay products, as shown in Table 15. The presence
of polonium alphaparticles in a RaaBe source is governed by the decay
of radium D, with a 22year halflife. Hence, in relatively young (a few
years old) sources, this last alphaparticle is not appreciably present.
TABLE 15
AlphaParticle Energies from Radium and Its Decay Products
AlphaEmitter
HalfLife
Energy (Mev)
Ra
1620 y
J 4.795 (93.5%)
14.611 (6.5%)
Rn
3.825 d
5.486
RaA
3.05 m
5.998
RaC
1.5 X 10 4 s
7.680
RaF(Po)
138.3 d
5.300
The rest of the alphaparticles are fully present after a few weeks. All
the following discussion (spectrum, yield) is concerned with such young
sources. Since both the spectrum and yield of PoaBe sources are
comparatively well known, it is easy to take into account the changes
in the source due to the accumulation of polonium.
The RaaBe neutron spectrum has not been nearly so extensively
investigated as PoaBe, mainly because of the difficulties of neutron
measurement in the accompanying high gammaray background. Its
properties may be roughly summarized as follows (A27): The fast
neutron spectrum extends to a maximum energy of ~13 Mev, with a
broad peak at ~4 Mev. There appears to be a substantial group of
intermediateenergy neutrons, but there is considerable uncertainty as
1 A large number of these sources were prepared in the laboratory of the Radium
Chemical Company, 570 Lexington Ave., New York City.
Sec. 3B] Sources and Detectors 365
to their amount and energy. Various estimates of the yield of inter
mediate neutrons range from ~1030 percent of the total yield. These
have been ascribed to the reaction
He 4 + Be 9 * 3He 4 + n 1
(in which a number of intermediate steps have been omitted). The
lowenergy group may also arise, in some part, from a (y,n) reaction on
beryllium. In any event, the lowenergy group does not seem to be
present in PoaBe sources, owing either to the absence of higherenergy
alphaparticles (those from RaC are assumed mainly responsible for
the 3a reaction) or of gammarays, or both.
Other light elements beside beryllium can be used to produce neutrons
through (a,n) reactions. Thus, a pressed RaaB source has been pre
pared, and its spectrum and yield studied (A27). Both B 10 and B 11
undergo exoergic (a,n) reactions (Q = 1.18 and 0.28 Mev, respectively),
the latter being responsible for most of the neutron yield. The spectrum
is comparatively simpler than that of a RaaBe source, rising rapidly to
a maximum at ~3 Mev, and then falling rapidly to zero at ~6 Mev.
There does not appear to be any appreciable intermediateenergy
component.
The reaction F 19 (a,n) is believed (from mass values) to be slightly
exoergic, by <0.5 Mev. Bretscher, Cook, Martin, and Wilkinson
(B62) have prepared a source composed of the relatively stable com
plex, RaBeF 4 , which they suggest for a standard neutron source, since
the characteristics of the complex are not expected to change appre
ciably with time.
The yields of the various sources discussed above have been studied
by a number of methods (A27). In general, for a given mixture of
TABLE 16
Yields or Radioactive (a,n) Sources fob Intimate Mixtures of
an AlphaEmitter and a NeutronProducing Material
Source Y (10 6 neutrons/curie • sec)
RaaBe 17
RnaBe 15
RaaB 6.8
PoaBe 3
RaBeF* 2.53 f
t Since the mixture is fixed, the value given is that of an actual source com
posed of the complex. This yield could be improved by adding beryllium, but
this would nullify the purpose of such a source, namely, the elimination of pos
sible changes due to alteration of the physical composition.
366
The Neutron
[Pt. VII
some alphaemitting compound (amat) and neutronproducing material
(X), the yield is given by the relationship
Y=Y
M(X)
M(X) + M(amat)
(83)
Values of F (yield for an <x> ratio of X to amat) are given in Table 16.
(<x,ri) yields from various materials have been extensively investi
gated, and are summarized by Anderson (A27). A number of these in
vestigations have employed thin sources and targets (S61, W3, H9),
while others have measured thick target yields. Most of the investi
gators employed polonium alphaparticles, varying their energy by
changing the pressure of gas between source and detector. The thin
target (a,n) cross section for beryllium (due to Halpern, H9) is shown
in Fig. 28a. Figure 28b shows the results of Segre and Wiegand on the
thick target yields of beryllium, boron, and fluorine. Thick target
yields, for (artificially accelerated) 9Mev alphaparticles, have been
measured by Ridenour and Henderson (Rll), and for 30Mev alpha
TABLE 17
Neutron Yields fob Polonium AlphaParticles on Thick Targets
Yield per 10 6 Alphas
Element
Roberts
Segre and
Wiegand
(A27)
Walker
Halpern
Szalay
(A27)
(W3)
(H9)
(S61)
Li
2.6
4.7
Be
80
73
50
B
24
19
19
C
0.11
N
0.01
0.07
F
12
10
Na
1.5
Mg
1.4
0.5
Al
0.74
0.25
0.22
Si
0.16
CI
0.11
A
0.38
Sec. 3B]
Sources and Detectors
367
0.2
0.6
1.0
a range in air (cm)
1.4 1.8 2.2 2.6
3.0
3.4
3.8
0.44
0.40
0.36
"£•0.32
a
a 0.28
x>
"^0.24
o
'•§ 0.20
o>
So.16
CO
J 0.121
0.08
0.04
I I j 1
 Be
' 1 ■] ! 1
l ■ i v — 1
i  ( I "

• •M
• •
•
•
•
•
•
• •
•
•
•
•

•
• •
•
•
•
•
•
 •
 ••)• i i
1 1
1
1 1 1
1
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
Energy (Mev)
(a)
4.5
5.0
80
70
Jso
a,
J 50
fl
1 Beryllium
2 Boron
3 Fluorine
'2
3
2 3
Energy (Mev)
(b)
Fig. 28. (a,n) yields vs. neutron energy for a number of elements, (a) Thin target
(a,n) cross section of beryllium, due to Halpern (H9). (b) Thick target (a,n) excita
tion functions of beryllium, boron, and fluorine, due to Segre and Wiegand (A27).
368 The Neutron [Pt. VII
particles by Allen, Nechaj, Sun, and Jennings (A8), who also measured
the angular distributions of the neutrons. The results of a number of
investigations of thick target yields from various elements for polonium
alphaparticles are summarized in Table 17, in which the data of Rob
erts and of Segre and Wiegand are taken from Anderson's summary
(A27). The results of the last three investigators are obtained by inte
gration of their thin target yield curves.
2. Photoneutron Sources. Photons can cause neutron emission from
nuclei, provided that their energy is greater than the neutron binding
energy. In so far as the (y,n) reaction is concerned, the periodic table
is conveniently divided into two groups: (1) deuterium and beryllium,
and (2) all the rest. The former have (y,n) thresholds of (2.226 ± 0.003)
and (1.666 ± 0.002) Mev, respectively. The latter have thresholds in
excess of 6 Mev. Since no radioactive nuclei which emit gammarays of
such high energy are known, radioactive (y,n) sources employ only
beryllium or deuterium. The (prompt) gammarays from various nu
clear reactions or the xrays obtainable from highenergy electron
accelerators can be used to obtain neutrons from other nuclei as well.
(a) Radioactive (y,n) Sources with Deuterium and Beryllium. The
bombardment of deuterium or beryllium by monoenergetic gammarays
results, for a given angle 6 between the emitted neutron and incident
gammaray, in monoenergetic neutrons, according to the relation
ship (W8)
B '  (^T 2 )
ET 2
E y Q
5^E y
1862(A  1)
2(A  l)(E y  Q)
931 A 3
+ 5 cos 6 (84a)
(84b)
E n is the neutron energy; E y , the gammaray energy; Q, the neutron
binding energy, all in Mev. A is the mass number of the target nucleus.
In almost all practical photoneutron sources, for reasons of intensity,
the gammaray source is surrounded by beryllium or deuterium, so that
the neutrons have an inherent energy spread corresponding to an iso
tropic distribution in the angle 0,
AE n = 25 (84c)
For 100kev neutrons, AE n /E n « 4 percent for beryllium, and « 25
percent for deuterium. The relative spread decreases with increasing
neutron energy.
The above is, however, not the main cause of energy spread. For
most sources, a larger uncertainty in the neutron energy arises from the
Sec. 3B]
Sources and Detectors
369
fact that considerable quantities of beryllium or deuterium must be used
to obtain usable neutron intensities. Since both beryllium and deuter
ium are quite light, and since the neutrons have a nonnegligible proba
bility of undergoing a scattering before emerging from the source, an
appreciable energy spread may thereby be introduced. In addition,
neutron scattering in the source has the effect of reducing the mean
energy of the emerging neutrons. Furthermore, the gammarays can
lose energy by Compton scattering and then produce neutrons, an
effect which introduces a further uncertainty in the neutron energy.
Since the Compton cross section is ~10 3 times that for photodisinte
gration, a source containing an infinite quantity of beryllium or deuter
ium will produce only ~1 neutron per 1000 gammarays.
TABLE 18
Photoneutbon Soukces
Standard
Yield 
Source
Tl A
£ T (Mev)
E n (Mev)
(10 4 neutrons/
sec • curie)
(1 gram at
1 em)
Na 24 + Be
14. 8h
2.76
0.83
13
Na 24 + D 2
2.76
0.22
27
Mn 66 + Be
2.59h
1.81,2.13,2.7
0.15,0.30
2.9
Mn 66 + D 2
2.7
0.22
0.31
Ga 72 + Be
14. lh
1.87,2.21,2.51
(0.78) f
5
Ga 72 + D 2
2.51
0.13
6
Y 88 + Be
87 d
1.9,2.8
0.158 ± 0.005 §
10
Y 88 +D
2.8
(0.31)
0.3
In 116 + Be
54 m
1.8,2.1
0.30
0.82
Sb 124 + Be
60 d
1.7
0.024 ±0.003 §
19
La 140 + Be
40 d
2.50
0.62
0.3
La 140 + D 2
2.50
0.151 ± 0.008 §
0.8
MsTh + Be
6.7y
1.80, 2.62
0.827 ± 0.030 §
3.5
MsTh + D 2
2.62 (ThC")
0. 197 ± 0.010 §
9.5
Ra + Be
1620 y
1.69, 1.15,1.82,
2.09, 2.20, 0.4#
a mess
3.0
Ra + D 2
2.42
0.12
0.1
t The standard yield is taken to be that of 1 gram of beryllium or heavy water
at 1 cm from 1 curie of the substance indicated.
% Values in parentheses are estimates.
§ Due to Hanson (H33).
370 The Neutron [Pt. VII
These factors, as well as other aspects of the production and use of
photoneutron sources, are discussed in considerable detail by Watten
berg (W8,W7). Table 18 summarizes the properties of available radio
active (y,n) sources.
The characteristics of the cross section for the D(y,n) reaction have
been discussed in Section 1. Considerably less is known concerning the
Be(y,n) cross section. The available evidence, both experimental (R26)
Fig. 29. Design of the primary photoneutron source standard at the National
Bureau of Standards (C42). The beryllium sphere is 4 cm in diameter and holds,
at the center (C), a 1gram capsule of radium.
and theoretical (G32), indicates that the cross section passes through
at least one maximum and one minimum as the gammaray energy is
increased from threshold to 2.76 Mev.
Prior to the extensive availability of strong artificially produced
gammaray sources, RayBe sources were extensively used to provide
intermediateenergy neutrons. Following a suggestion of Gamerts
f elder and Goldhaber (G2), such a source has been prepared by Curtiss
and Carson (C42) at the National Bureau of Standards, to serve as a
permanent neutron standard. Since a Ra7Be source does not require
mixing of the radium and beryllium (with the attendant possibilities
for physical change), its neutron output should not vary with time. In
the standard source, a pressed radium bromide pellet is placed at the
center of a carefully machined sphere of beryllium metal; the design
of the standard source is shown in Fig. 29.
The absolute yield of such a source could be obtained, without the
necessity of any neutron measurement, by a method developed by
Sec. 3B] Sources and Detectors 371
Paneth and Gluckauf (P2, G8). They measure the total accumulation
of helium 1 after a known time of irradiation of the beryllium.
An interesting application of the photodisintegration process is as a
gammaray detector which is completely insensitive to gammarays of
energy below the photodisintegration threshold (P8a). Myers and Wat
tenberg (M42) have used this device to detect the presence of a small
component of "crossover" gammarays when two or more gammarays,
both of which are below the photodisintegration threshold of beryllium
or deuterium, are emitted in cascade.
(b) Photoneutrons from HighEnergy GammaRays and XRays. Pho
toneutron reactions with highenergy gammarays have been investi
gated in a large number of elements. Following the work of Bothe and
Gentner (B52), the ~17Mev Li(p,y) gammarays have been used to
study the (y,ri) cross sections of many elements, by Waffler and Hirzel
(Wl) and by McDaniel, Walker, and Stearns (M12). However, the
most extensive investigations have employed xray beams from electron
accelerators, mainly betatrons.
The xrays are produced by causing the accelerated electrons to strike
a target, usually of some heavy element. The resulting radiation is
allowed to fall on the material under investigation. (7,71) reactions are
detected either through direct observation of the neutrons, or by de
tection of the radiations from the product (usually /3 + radioactive)
nuclei.
The shape of the xray spectrum from an electron accelerator depends
on the target thickness. For relatively thin targets, the distribution of
xray quanta follows a bremsstrahlung spectrum, at least for energies
not too far below the maximum (electron) energy,
dN(E y ) = —1 (85)
Ey
The measurement of (y,n) cross sections with such xray beams involves
the complication of dealing with a heteroenergetic source. However, if
the electron energy can be varied, (y,n) yields can be measured as a func
tion of the maximum xray energy. The results of three such studies,
due to Diven and Almy (D15), are shown in Fig. 30a. Such curves can
be interpreted in terms of the (y,n) cross section vs. E y , provided that
the xray spectrum is known and the xray intensity (the value of k in
Eq. 85) is calibrated.
1 The reaction is Be 9 (7,n)Be 8 ; Be 8 > 2He 4 .
372
The Neutron
[Pt. VII
Some curves of <r(y,ri) vs. E y are shown in Fig. 30b. The striking fea
ture of such crosssection curves, first noted by Baldwin and Klaiber
(B7), is the strong resonance shape. This shape has been observed for
all the nuclei studied, although the positions of the maxima and the
16
14
12
za
a
g 8
B
a)
Ag 1
07,109
Cu 63
/A
J 27
10 12 14 16 18 20
Maximum xray energy (Mev) •
22
24
Fig. 30a. Relative neutron yields vs. maximum xray energy for three nuclei, due
to Diven and Almy (D15). The ordinates are in arbitrary units.
resonance widths vary from nucleus to nucleus (D15, M13). Particu
larly accurate work, determining the resonance constants for many
nuclei, has been done by the Saskatchewan Group (J4, K4).
An explanation of these resonances has been advanced by Goldhaber
and Teller (G16). They have postulated the possibility of dipole vibra
tions in which the protons (as a whole) oscillate with respect to the neu
trons in the nucleus. The general features of the process of dipole radia
tion capture have been derived by Levinger and Bethe (LI 6).
Sec. 3B]
Sources and Detectors
373
Neutron yields, due to the (y,n) reaction, have been measured for
many elements and at many xray energies, up to 330 Mev (P31, T8).
Neutron sources of considerable strength can be obtained, through
(y,n) reactions, from electron accelerators. Thus, by using the elec
trons from a 3.2Mev linear accelerator to produce gammarays in a
lead target, and by irradiating a heavy water (or beryllium) target with
these gammarays, Cockroft, Duckworth, and Merrison (C16) obtained
w
Ta
p\
3u 6
3
W 11 12 13 14 15 16 17 18 19 20 21
7 ray energy (Mev)
Fig. 30b. (y,n) cross sections vs. 7ray energy for copper, silver, and tantalum
(D15.M13).
average neutron yields of ~10 9 neutrons/sec and peak yields of ~2
X 10 12 neutrons/sec (in ~2 ^sec bursts). Calculations based on the
curves of Fig. 30 indicate that an electron beam of 1 /tamp (average)
and 20 Mev energy, impinging directly on a thick target of a medium
or heavy element, should yield ~10 10 neutrons/sec.
The spectrum of neutron energies resulting from xrayinduced (y,n)
reactions is relatively broad, corresponding to "evaporation" from an
excited compound nucleus; the theory of the evaporation process, and
of its Zdependence, has been discussed by Heidmann and Bethe (H52).
The Maxwellian form of the neutron energy distribution has been veri
fied by Byerly and Stephens (B80). However, there is expected to be a
highenergy "tail" of neutrons ejected by direct gammaray action, with
a nonspherical angular distribution (C29).
374 The Neutron [Pt. VII
3. Accelerated Charged Particle Sources. Radioactive alphaparticle
and gammaray neutron sources are of rather limited usefulness, espe
cially for the production of intense beams of monoenergetic neutrons.
The availability of Van de Graaff and cyclotron accelerators, capable
of delivering strong monoenergetic currents of various nuclear projec
tiles, makes possible the production of strong neutron sources by a
variety of nuclear reactions. These projectiles include protons (p),
deuterons (d), tritons (t), alphaparticles (a), and heavier nuclei. In
this section we shall consider only reactions induced by the first three.
[(a,n) reactions have been discussed in connection with radioactive
neutron sources.]
We shall mainly emphasize, in this section, reactions which can lead
to monoenergetic neutrons. This possibility exists whenever the energy
of the first excited state of the product nucleus is too great to be excited
by the projectiles used. Depending on the nucleus involved, the first
excited state may lie anywhere from a few hundred kev to many Mev
above the ground state. In some cases, sources may be effectively mono
energetic even when the first excited state is energetically available,
owing to a relatively low yield of excited product nuclei.
With the available reactions which yield monoenergetic neutrons, it
appears possible to cover the neutron energy range from a few kev to
20 Mev. To cover the rest of the very fast and the ultrafastneutron
ranges, it is necessary to resort to heteroenergetic sources, and to de
pend on the detector for any sorting out of the neutron energies. Prior
to the availability of tritium, and the use of the td reaction, heteroener
getic (d,n) reactions were the only ones available for obtaining very
fast neutrons. Such sources will also be considered in this discussion.
Given the requisite reaction, a strong current of monoenergetic
projectiles, and a thin target, the neutrons emerging at a given angle
with respect to the projectile direction have a definite energy. The rela
tionship between the neutron energy E n , the angle 8, the projectile
energy E i} the reaction Q value, and the masses of the particles involved
has been frequently described. 1 The properties of monoenergetic neu
tron sources are collected in a review article by Hanson, Taschek, and
Williams (H34), upon which this summary leans heavily. The energy
angle relationships in a given reaction are conveniently represented in
the form of a nomograph, developed by McKibben (M15), 2 of which
some examples will be shown in the following.
1 See, for example, Part VI.
2 Such nomographs for the most extensively used reactions can be purchased as
document MDDC 223 from the Document Division of the AEC, Oak Ridge, Ten
nessee. They have been reprinted in the article of Hanson, Taschek, and Williams
(H34).
Sec. 3B] Sources and Detectors 375
A McKibben nomograph (Figs. 31, 33, 37, 40) consists of two sets of
semicircles (solid and broken) and two sets of radial lines. The solid
semicircles, centered at the origin of the (lower) neutron energy scale,
represent various values of the neutron energy in the laboratory coordi
nate system. The solid radial lines, from the same origin, represent
angles of emission of the neutrons in the laboratory system. The
broken semicircles (which are not concentric) represent various energies
of bombarding particle (p or d) ; the bombarding energy is given by the
intersection of a broken semicircle with the upper of the two horizontal
axes. The broken radial lines represent loci of equal values of the angle
of neutron emission in the centerofmass coordinate system.
Thus, given a value of the bombarding particle energy, the energies
of the emitted neutrons are given by the intersections of the appropriate
broken semicircle and the solid semicircles; to each neutron energy (in
tersection) there corresponds a definite laboratory angle (solid line) and
a definite centerofmass angle (broken line).
(a) Intermediate and Fast Neutrons from (p,n) Reactions. These reac
tions are endoergic. Hence, by bombarding thin targets with protons
of energy only slightly above threshold it is possible to obtain monoener
getic neutrons of relatively low energy. The minimum energy obtain
able from such sources is limited by the fact that, at proton energies
only slightly above threshold, there are two neutron energies correspond
ing to each angle of emergence. (Neutrons emitted at forward and back
ward angles in the centerofmass system all lie within a cone of apex
angle <180° in the laboratory system.) However, as soon as the apex
angle of the cone of neutron emergence becomes 180°, the energyangle
relationship is unique. In this respect, the heavier the target nucleus,
the lower the energy at which the neutrons for a given angle are mono
energetic. However, the necessity for penetration, by the proton, of
the Coulomb barrier limits the possible target nuclei to low Z (<25).
Some properties of known {p,n) reactions are summarized in Table 19.
Most of these data are from the work of Richards and coworkers at the
University of Wisconsin, and have been collected by Richards, Smith,
and Browne (RIO), who give references to the original investigations.
The minimum neutron energy at threshold (fifth column) arises from
the centerofmass motion of the system.
E t
E  mi ° = CA+I) 5 (86)
where A is the mass number of the target nucleus. The sixth column
gives the minimum energy of monoenergetic neutrons in the forward
376
The Neutron
[Pt. VII
TABLE 19
Properties op (p,n) Reactions for Z < 25 (Mostly from RIO)
Target
Product
Observed
E, (Mev)
Q
(Mev)
■^w.min
(kev)
^ re.min
at0°
(kev)
Lowest
Level
(Mev)
xH 2
21H 1
3.339 ±0.015
2.225
371
1979
iH«
2 He 3
1.019 ±0.001
0.764
63.7
286.5
>2.5
3 Li 7
4 Be 7
1.882 ±0.002
1.646
29.4
120.1
0.435
4 Be 9
5B 9
2.059 ±0.002
1.852
20.6
83.4
>1.5
6 B"
eC 11
3.015 ±0.003
2.762
20.9
84.5
2.02
6 C 12
6 N 12
20.0 ±0.1
18.5
118
477
eC 13
6 N 13
3.236 ± 0.003
3.003
16.5
66.4
2.383
6 C 14
6 N"
0.664 ±0.009
0.620
2.9
11.8
2.3
8 18
8 F 18
2.590 ±0.004
2.453
7.2
28.8
9 F 19
ioNe 19
4.18 ±0.25
3.97
10.5
42
nNa 23
i 2 Mg 23
4.78 ±0.3
4.58
8.3
33
itCI"
isA 37
1.640 ±0.004
1.598
1.1
4.6
1.4
18 A«>
19K*
<2.4(?)
2.3(?)
1.6
5.7
0.81
i»K«
2oCa 41
1.25 ±0.02
1.22
0.7
2.8
1.95
2lSc 45
22 Ti 45
~2.85
2.79
1.35
5.4
2 3 V 61
2 4Cr»
1.562 ±0.006
1.532
0.58
2.3
0.775 f
25 Mn 55
26 Fe 66
1.18 ±0.01
1.16
0.38
1.5
1.020 ±0.010
1.00
0.33
1.3
0.42 %
t (S52). t (S54, M10).
direction, E' ntm  m , which corresponds to the forward cone just filling the
forward hemisphere. The properties of the most important of these
neutron sources follow:
H 3 (p,n)He 3
McKibben's nomograph for this reaction is shown in Fig. 31. Since
tritium has only recently become extensively available, this reaction
has not been very widely used for a neutron source; only the Wisconsin
group has reported extensive (cross section) measurements with this
source (B41, M31). Most of the information concerning yields, angular
distributions, etc., vs. proton energies up to 2.5 Mev is due to the Los
Alamos group (J2, H34). The cross section is shown in Fig. 32. The
rapid increase may be due to a resonance of He 4 corresponding to in
cident protons of >2.5 Mev energy. The angular distribution of the
emerging neutrons is quite complex (J2). A large yield of 20Mev
T(p,re)He 3
3.0
180^
E n (Mev)
Fig. 31. McKibben nomograph of energy relationships in the H 3 (p,n) He 3 reaction
(H34). Directions for its use are given on p. 375.
'o
e
J 30
*?
0.10
1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4
•Ep(Mev)
Fig. 32. Total cross section of the H 3 (p,n)He 3 reaction as a function of the incident
proton energy {32).
377
378
The Neutron
[Pt. VII
4.0 3,5 3.0 2.5
,iiiii — r
1.0 0.5
2.0 E p 1.88 (Mev) 2.0
3.0 3.5 4.0
! I I I i I I M I I  i I I ' I"
0.5 1.0 1.5 2.0
E n (MeW
Fig. 33. McKibben nomograph for the Li 7 (p,ra)Be 7 reaction (H34). Directions for
its use are given on p. 375.
11
10

I
/
\ F40kev Li target
4
'
J
■
/
■
.
'
7
''E n (0°)
'
'
\
.
/
\
/
/
4
1
/
\
j
/
\
/
[/_
}
J
/
1/
1
}
f
/
2.0
s
15 X
1.0
0.5
1.8 2.0 2.2 2.4 2.G 2.8 3.0 3.2 3.4 3.6
Proton energy (Mev)
Fig. 34. Yield of the Li 7 (p,«)Be 7 reaction in the forward direction (H34).
Sec. 3B]
Sources and Detectors
379
gammaradiation, arising from the H 3 (p,7)He 4 reaction, accompanies
the (p,n) reaction, and may interfere with some forms of neutron detec
tion (A30).
Li 7 ( P ,n)Be 7
This reaction has provided the most extensively used source of inter
mediateenergy neutrons. Its nomograph is shown in Fig. 33. The
S 0.30
0.20
L
i 7 (p,
«)B
3 7 cross section as a function of
energry'

2 Points from Be 7 activity
^1 Calibration points from Mn 55 bath standardizatio
o Points from integrated angular distributions
i
Foir
its I
rom
E1SS1C
n ra
tes
t\
/ :
:
\ c
Y
:
\ c
1
A
i
J (
t
. '
■> ■*
J
— ~»— — Extrapolation to threshold assumed linear in E„
£n£±£* Extrapolation to threshold assumed linear vas/E^
i
i
1.9 2.0
2.1 2.2 2.3
Proton energy (Mev)
2.4
Fig. 35. Total cross section for the Li 7 (p,ra)Be 7 reaction (H34).
yield in the forward direction is shown in Fig. 34. The first peak is due
to the concentration of neutrons in the forward cone for proton energies
close to threshold. The second peak, at E p « 2.3 Mev, is real, and
corresponds to a resonance of the compound nucleus Be 8 . The angular
distribution is peaked in the forward direction, especially in the vicinity
of the resonance, and requires terms up to cos 2 (P 2 ) for its description
(Tl). The total cross section vs. proton energy is shown in Fig. 35.
380 The Neutron [Pt. VII
Until relatively recently, it was thought that the neutrons from this
reaction were truly monoenergetic (H8). However, the reaction is now
known to yield a lowenergy group of neutrons corresponding to an ex
cited state of Be 7 at 435 kev. The results of various investigators on the
relative yield of the lowenergy group are summarized in Table 20.
TABLE 20
lTivb Yield of
Neutrons Arising from Excitation of the 435]
State
OF Be 7 IN THE
Li 7 (p,m)Be 7 * Reaction
Intensity Relative
Proton Energy
Angle of
to GroundState
(Mev)
Observation
Group (%)
References
2.378 (E t *)
0°
~3
W33
2.705
0°
8±2
H20
2.75
30°
9± 1.5
J5
2.89
30°
10.5 ± 1
J5
3.120
0°
8±2
H20
3.31
0°
17 ± 10
J8
3.49
0°
10 ±3
F42
3.66
30°
12 ± 1
J5
3.91
0°
9±4
J8
60°
16 ±6
J8
Additional Be 7 levels have been reported below 1 Mev (G30), but these
have not been confirmed in other investigations (K6).
Other Possibilities
Other possible (p,n) reactions are discussed by Hanson, Taschek, and
Williams (H34) and by Richards, Smith, and Browne (R10). The
Be 9 (p,n)B 9 reaction is quite similar to that on Li 7 , but does not give a
lowenergy group up to neutron energies of ~1.5 Mev (J9). However,
its yield is lower, and thin beryllium targets are considerably more
difficult to prepare.
The (p,n) reactions on scandium, vanadium, and manganese are
being considered for producing monoenergetic neutrons in the ~lkev
energy range. Their yields have been studied by a number of investiga
tors (H34, R10, B4, M10, S52, S54).
(6) Fast and Very Fast Neutrons from (d,ri) Reactions. These reac
tions are, with the exception of C 12 (d,n)N 13 , exoergic. They are useful
in producing fast neutrons in the range 120 Mev. The Q values for the
most useful (d,n) reactions are shown in Table 21. The last two yield
heteroenergetic neutrons, but have been extensively used because of
their high yields.
Sec. 3B]
Sources and Detectors
381
TABLE 21
Properties op Some (d,n) Reactions
Reaction Q Value (Mev)
H 2 (d,re)He 3 3.265 ±0.018 (T16)
H 3 (rf,w)He 4 17.6 (H34)
C 12 (d,w)N 13 0.281 ± 0.003 (T16)
N 14 (d,n)0 15 5.1 (H34)
Li 7 (d,n)Be s 15.0 (T16)
Be 9 (d,n)B 10 3.79 (T16)
E n0 for
E d = (Mev)
2.45
14.1
(E, = 0.328)
4.8
13.3
3.44
The last column gives the energy E n0 of neutrons which would result
from the absorption of zero energy deuterons; this energy is independent
1 2 3 4 5 6 7
Energy of accelerated particle (Mev)
Fig. 36. Energies of neutrons, emitted at 0° and 180°, for the most important mono
energetic (d,n) and (p,ri) reactions (H34).
of the angle of emergence of the neutron. For deuterons with a given
kinetic energy, the neutron energy is uniquely determined by its angle
of emergence, provided that the reaction goes to the ground state of the
product nucleus. The neutron energies vary on both sides of E n0 , being
greater for emission in the forward direction and less for backward
emission. The energies of neutrons emitted at 0° and 180° are shown in
Fig. 36, as a function of the bombarding energy, for the first three reac
382 The Neutron [Pt. VII
tions in the table. Also included are curves for the two (p,ri) reactions
discussed above.
H 2 (d,n)He 3
The dd reaction has been extensively used, and its characteristics
widely investigated (H34). It has high yields for low bombarding
energies, which makes it particularly useful as a neutron source with
lowvoltage (<1 Mev) particle accelerators. The neutrons are mono
energetic up to bombarding energies of 10 Mev. Figure 37 is the Mc
Kibben nomograph for this reaction.
EV(Mev) 2 3
Fig. 37. McKibben nomograph for the dd reaction (H34). Directions for its use
are given on p. 375.
The angular distribution of dd neutrons has been extensively studied,
both experimentally and theoretically. At energies of a few hundred
kev, the distribution in the cm. system is symmetrical about 90°, being
~60 percent lower at 90° than at 0° and 180° (H34, M16). For higher
energies the distribution becomes much more anisotropic. Hunter and
Richards (H78) have investigated the angular distribution for deuterons
of from 0.5 to 3.7 Mev. They find that it can be represented by an ex
pression of the form
N(fi) = A P (d) + A 2 P 2 {6) + AiP^e) + A 6 P 6 (d) +■■■ (87)
in which the P„(0)'s are the Legendre polynomials, and all the coeffi
cients vary with energy, as shown in Fig. 38a. The theory of the
angular distribution has been discussed by Konopinski and Teller (K21)
Sec. 3B]
Sources and Detectors
383
and by Beiduk, Pruett, and Konopinski (B14). The total cross section
vs. deuteron energy is shown in Fig. 38b.
E d (Mev)
Fig. 38a. Variation with deuteron energy, E d , of the coefficients of the Legendre
polynomial fit to the angular distribution of the neutrons from the dd reaction in
the centerofmass coordinate system (H78).
'
' '
i I r
0.10
o
^S^~* " "° '
"^ s— *
•
^0.08
D /
Symbol on
Reference No.
01
o
Figure
in Legend
S, 0.06
to
 o* .
•
Q
1
2
3
0.04
 &
.8
A
4
5
0.02
a
 +
E d (Mev)
(1) Hunter and Richards (H78). (2) Blair, Freier, Lampi, Sleator, and Williams, Phys. Rev., 74,
1599 (1948). (3) Manley, Coon, and Graves, Phys. Rev., 70, 101(A) (1946). (4) Graves, Graves,
Coon, and Manley, Phys. Rev., 70, 101(A) (1946). (5) Bretscher, French, and Seidl, Phys. Rev.,
73, 815 (1948).
Fig. 38b. Total cross section for the H 2 (d,n)He 3 reaction (H78).
The dd reaction is frequently used to provide strong neutron sources
by bombarding thick targets of heavy ice with deuterons. The thick
target yield curve (H34) is given in Fig. 39.
384
The Neutron
[Pt. VII
The companion reaction H 2 (d,p)K 3 has a comparable cross section
(M16, H34). The protons are frequently used to "monitor" the neutron
yield. At high bombarding deuteron energies the He 3 recoils from the
1
/2
10 'o
2 5
0.2
0.4 0.6 0.8
Deuteron energy (Mev)
1.0
Fig. 39. Thick heavy ice target H 2 (d,ra)He 3 neutron yields (H34).
(d,n) reaction can be used as a monitor, and also, by observing He 3
neutron coincidences, to eliminate background effects due to (d,ri) re
actions on target impurities.
H 3 (d,w)He 4
This, the td reaction, is the most strongly exoergic of the reactions
capable of yielding monoenergetic neutrons; it can provide neutrons of
from 12 to 20 Mev by using deuterons of up to 3 Mev (Fig. 36). The
McKibben nomograph for the reaction is shown in Fig. 40. This reac
tion is especially useful with lowvoltage deuteron accelerators, since it
has very large yields for deuterons of a few hundred kev energy, owing to
a resonance in the cross section at ~100 kev. The lowenergy cross
section data (up to ~1 Mev deuterons) can be fitted by the expres
sion (H34)
58 exp (1.72/2^)
a = — — — — — barns (88)
E 1 + {E  0.096)7(0. 174)'
where E is the deuteron energy in Mev. From 1 to 2.5 Mev the cross
section remains essentially flat at ~0.15 barn. The cross section is 0.05
barn at 10.5 Mev (B72).
Sec. 3B]
Sources and Detectors
385
The angular distribution in the td reaction, for incident deuterons up
to 2.5 Mev, has been measured by Taschek, Hemmendinger, and Jarvis
(T2) by observing the distribution of recoil alphaparticles. Their re
T(d,m)He 4
10 20
E {Mev) 10
Pig. 40. MeKibben nomograph for the td reaction (H34). Directions for its use
are given on p. 375.
suits are shown in Fig. 41. Detection of the recoil alphas provides an
effective means for monitoring the neutron yield from this source. The
angular distribution for E d = 10.5 Mev has been measured by Brolley,
Fowler, and Stovall (B72).
~0.016
"3
* 0.012
"0.008
^^ T
fflEJ
2.5  ^4A°;f<4.2.5
1.5
—  ±
! 0.024
40 50 60 70 80 90 100 110 120 130
Laboratory angle (degress)
30 60 90 120 150
Center of Gravity angle (degrees)
Fig. 41. Angular distribution of recoil alphaparticles from the H 3 (<2,n)He 4 reac
tion for various deuteron energies (in Mev) (T2, H33).
386 The Neutron [Pt. VII
Because of the strong lowenergy resonance, the td reaction is expected
to have large thick target yields. Thus, for a thick gas target and 600
kev d's the yield is 5 X 10 8 neutrons per microcoulomb ; for tritons ad
sorbed in a thick zirconium target, the yield at Ed = 200 kev is about
10 8 neutrons per microcoulomb (H34).
One interesting application of the td reaction is in the possibility of
converting intense sources of thermal neutrons (such as those available
from piles) into strong fastneutron sources. Thus, if thermal neutrons
are allowed to fall on a mixture of lithium and deuterium, fast tritons
(E = 2.65 Mev) will be produced by the Li 6 (n,a)H 3 reaction. These
tritons then react with the deuterons to produce 14Mev neutrons by
the td reaction. A recent investigation by Almqvist (All) has shown
that such sources also produce neutrons by reactions of tritons on the
lithium nuclei; the latter reactions have a considerably greater yield
than the td reaction in the mixtures used. Since the neutrons from
lithium result from a variety of reactions on both lithium isotopes, with
Q values ranging from 16 to 8.9 Mev, the neutrons from such sources
have a complex spectrum. The neutron yields for a number of com
pounds are shown in Table 22. From these data the author derives
TABLE 22
Neutron Yields fkom Thermal Neutron Irradiation of Lithium and
LithiumDeuterium Compounds (All)
Neutrons per 10 6 Tritons
Substance
From Li
From '.
LiF
26.8
Li 2 C0 3
17.2
LiOHH 2
15.2
LiOD
20.3
3.4
LiODD 2 + D 2
9.3
12.1
average cross sections of 1.5 and 0.41 barns for 2.65Mev tritons on
thick targets of lithium and deuterium, respectively. Crews (C32) has
measured the yield and angular distribution of neutrons from thin
lithium targets for triton energies between 0.25 and 2.10 Mev; he
derives a ~ 0.76 barn for E t = 2.0 Mev on a thick Li target.
C 12 (rf,n)N 13
This, among the (d,n) reactions of interest, is the only endoergic
reaction. It is mainly useful because of its low threshold (Table 21;
Sec. 3B] Sources and Detectors 387
Fig. 36), which makes it a convenient lowvoltage accelerator source for
neutrons of from a few hundred kev to ~1.5 Mev. Since the first ex
cited state of N 13 , observed in this reaction, is at ~2.3 Mev (G29), the
neutrons are monoenergetic to —2 Mev. The reaction yield is rela
tively low and shows a complex angular distribution and energy de
pendence (H34).
The companion reaction C 12 (d,p)C 13 has a similar yield curve at low
deuteron energies, and is useful for monitoring the neutron yield. A
complicating feature is the presence, with normal carbon targets, of the
reaction C 13 (d,n)N 14 , with a Q value of 5.2 Mev, which gives a fast
neutron group of intensity <~1 per cent of the C 12 neutrons.
N 14 (<Z,n)0 15
This reaction, very seldom used, gives monoenergetic neutrons for
deuterons of up to ~1 Mev energy. At 1 Mev the cross section relative
to that of the dd reaction has been measured by Gibson and Livesey
(G4), and is shown in Table 23.
TABLE 23
Ratio of dd to N 14 rf Cross Sections for 1Mev Deuterons (G4)
Angle Ratio (dd/N u d)
0° 4.4
30° 4.4
90° 1.5
150° 2.1
U 7 (d,n)Be 8
Because of its large Q value and large yield, this source has been exten
sively used, especially with cyclotrons. However, the large number of
levels, in both the compound and product nuclei, gives rise to a com
plicated neutron spectrum with a wide energy spread (H34). The thick
target yield for a bombarding energy of 600 kev is 17 X 10 6 neutrons
per microcoulomb. Above E d = 1 Mev the thick target yields are
greater than for the dd reaction.
Be 9 (d,w)B 10
This reaction also gives rise to a complex neutron spectrum of lower
maximum energy than that from Li 7 (A4). The thick target yields at
388
The Neutron
[Pt. VII
low deuteron energies are shown in Table 24. Above ~1 Mev the thick
target yields are between those of the dd and the Li 7 (d,ri) reactions.
TABLE 24
Thick Taeget Yields fob the Be 9 (d,w)B 10 Reaction
Ed (kev) 10 6 Neutrons/Microcoulomb
400
4
600
21
800
106
Other Possibilities
Many other (d,n) reactions have been employed as neutron sources.
Among these the F 19 (d,ra)Ne 20 , Q = 10.7 Mev, has received some atten
tion, but its characteristics are not more favorable than the other reac
tions which yield heteroenergetic sources, and its yield is lower. While
such sources have frequently been used in the past to produce very fast
neutrons, they have been mainly superseded by the td reaction.
The (d,ri) yields of a variety of elements, for deuteron energies of
~10 Mev and greater, have been studied by a number of investigators.
Thick target yields for 10Mev deuterons (due to Smith and Kruger,
S41) are shown in Table 25, together with the yields at 15 Mev (due to
Allen, Nechaj, Sun, and Jennings, A8). The latter results are sur
TABLE 25
Neutron Yields from 10 and 15Mev Deuterons on Thick Targets
Yield (10 8 neutrons/
Yield (10 s neutrons/
microcoulomb)
microcoulomb)
Target
Z
Target
Z
10 Mev
15 Mev
10 Mev
15 Mev
Be
4
320
190
Nb
41
15
B
5
190
Mo
42
42
15
C
6
120
Ag
47
14
Al
13
87
64
Cd
48
12
P
15
105
Sb
51
35
Ti
22
65
Ta
73
7.4
3.3
Cr
24
29
W
74
7.0
Mn
25
76
52
Pt
78
6.0
Co
27
26
Au
79
4.7
2.1
Ni
28
33
Pb
82
2.1
Cu
29
55
29
Bi
83
1.3
Sec. 3B] Sources and Detectors 389
prisingly lower than those at 10 Mev, although the trend with atomic
number is the same for both series of measurements. The discrepancy
probably arises, at least in part, from the fact that the 10Mev measure
ments counted neutrons of all energies while the 15Mev results apply
only to those neutrons capable of exciting the S 32 (n,p) reaction.
The angular distributions of (d,n) neutrons have been investigated by
Roberts and Abelson (R14), by Falk, Creutz, and Seitz (Fl), by Allen
et al. (A8, S58) (all at E d = 15 Mev), by Ammiraju (A22) {E d = 18
Mev), by Schecter (S4) {E d = 20 Mev), and by others. These distri
butions show a strong peaking in the forward direction. [Some, e.g.,
Be, when only the highestenergy neutrons are detected, show structure
in the angular distribution (S58) .]
The observed characteristics of the (d,ri) reaction for highenergy
deuterons can be explained in terms of the "stripping" theory of Serber
(S20) and Peaslee (P8), according to which most of the neutrons are
produced by a process in which the neutron never enters a compound
nucleus; the deuteron is polarized in the field of the nucleus and then
split, the proton being captured (and occasionally scattered) by the tar
get nucleus. Recently, Butler (B78, B79) has made a significant con
tribution to the "stripping" theory by considering in greater detail the
angular distribution of neutrons from such reactions when the product
nucleus is left in a definite quantum state. His calculations predict
structure (i.e., maxima and minima) in the angular distribution which
can be interpreted in terms of the angular momentum and parity
properties of the initial and final states.
(c) Ultrafast Neutrons from Accelerated Deuterons and Protons. When
charged particles from ultrahighenergy accelerators strike a target,
neutrons are produced through a variety of reactions. Those reactions
which involve the capture of the bombarding particle into a compound
nucleus give rise to neutrons by the process of evaporation. Such neu
trons have the broad energy distribution characteristic of the statistical
theory, with the maximum in the fast or very fastenergy region and a
moreorless symmetrical angular distribution. However, when deu
terons (and, to a lesser extent, protons) are used as the bombarding
particles, there is observed, superimposed on the evaporated neutrons,
a strong forward peak of ultrafast neutrons with a relatively narrow
energy distribution. In the case of deuteron bombardment, the neu
trons arise mainly from the stripping process, mentioned briefly in the
preceding section. Serber (S20) has given a simple picture of this
process, as follows:
The deuteron is a relatively loosely bound structure in which neutron
and proton spend a large fraction of the time far apart. Hence, it is
390 The Neutron [Pt. VII
not improbable that, as a deuteron traverses the target, the proton will
strike one of the target nuclei while the neutron remains outside the
nucleus. In such a collision the proton will be stopped while the neu
tron will go on, carrying off approximately half of the original deuteron
energy.
However, the neutron has internal motion, relative to the center of
mass of the deuteron. Hence, as the neutron leaves the proton behind,
this relative motion is superimposed on the forward motion of the
deuteron, giving rise to an energy spread centered around the average
value of half the deuteron energy. Another result of the relative motion
is that the emergent "stripped" neutrons have a spread in directions,
around the original deuteron direction.
The magnitudes of the energy and angular spreads of the neutron
beam can be estimated as follows: The forward momentum of the
neutron, due to the kinetic energy, E d , of the deuteron, is
Po = (ME d ) 1A (89a)
The internal momentum of the neutron is, roughly,
Vi ^ (Me d ) y ° (89b)
where t d is the deuteron binding energy. The stripped neutrons have
energies within the limits given by
(po ± Pi) 2 1
E ^ n P ~2 Ed± {tdEd) (90)
A more accurate calculation (S20) gives, for the full width at half
maximum,
AE = 1.5(tdE d ) H (90a)
The angular spread in the neutron beam is determined by the relative
values of the forward and transverse neutron momenta:
A0 « =* 2—1 radians (91)
Po \eJ
The more accurate calculation (S20) gives
A0 = 1.6 () ' (91a)
as the full angular width at half maximum. The above considerations,
with respect to the angular spread, apply only to targets of light nuclei,
in which the coulomb deflection of the deuterons is negligible. For the
Sec. 3B]
Sources and Detectors
391
heaviest nuclei the coulomb deflections lead to about twice as large a
spread as Eq. (91a).
The energy and angular distributions of the stripped neutrons are
superimposed on the background of neutrons produced as a result of
evaporations. However, this background is expected to be small,
especially since it is spread, approximately uniformly, over all direc
tions. In addition, some neutrons are produced by the disintegration
of deuterons in the coulomb field of the target nuclei; this effect is rela
tively small for light nuclei and becomes comparable to the stripping
production for the heaviest target materials (Dl, K20).
40 60 80 100
Neutron energy (Mev)
120
140
Pig. 42. Energy distribution of neutrons obtained by stripping 190Mev deuterons
in a 1.27cm thick beryllium target, due to Hadley et al. (HI). The curve is from
the theory of Serber (S20).
The observations of Helmholz, McMillan, and Sewell (H53) on
angular distributions of the neutrons produced by bombarding various
nuclei with 190Mev deuterons and of Hadley, Kelly, Leith, Segre,
Wiegand, and York (HI) on the energy distribution of the neutrons
from a beryllium target bombarded by 190Mev deuterons are in good
agreement with Serber's theory, and demonstrate the possibility of
using the stripping process to obtain relatively monoenergetic ultra
fast neutron beams from ultrahighenergy particle accelerators. The
energy distribution due to the latter investigators is shown in Fig. 42.
When ultrafast protons are used as the bombarding particles, the
emerging neutrons are neither so sharply collimated nor so nearly mono
energetic. However, observations by Miller, Sewell, and Wright (M32)
(330Mev protons on beryllium, aluminum, copper, and uranium) and
by DeJuren (Dll) (340Mev protons on beryllium) have demonstrated
that there is an appreciable component of ultrafast neutrons emitted at
forward angles. The results of the firstmentioned investigators, on the
full angular widths at half maximum for neutrons of E > 20 Mev, are:
beryllium, 54°; aluminum, 59°; copper, 59°; uranium, 58°.
392 The Neutron [Pt. VII
Such forwardcollimated, ultrafast neutrons result primarily from one
or a few pn collisions in the nucleus ; at these energies the scattering of
neutrons by protons is predominantly forward (charge exchange), and
the scattered neutron has an appreciable probability of emerging from
the nucleus without undergoing any further scattering.
Relative neutron yields for targets of various Z have been measured
by Knox (K20) with both highenergy deuterons and protons as the
bombarding particles. For 190Mev deuteron bombardment the rela
tive yields agree well with stripping plus disintegration by the Coulomb
field. In the case of 340Mev proton bombardment, the yield varies
approximately as (A — Zf 3 .
The energy distribution of the neutrons from beryllium and carbon
targets bombarded with ~100Mev protons has been investigated by
Bodansky and Ramsey (B42, B43). For a beryllium target they ob
serve a sharp peak at E n « 93 Mev, with a width at half maximum of
~30 Mev. Below ~70 Mev, the neutron energy distribution becomes
essentially fiat, with a yield of ~40 percent of the peak value. The neu
tron yield from a carbon target shows a slight peak at ~70 Mev (~25
percent above the roughly constant yield below E n = 60 Mev) and
falls off rapidly above 70 Mev. Similar studies on neutrons from beryl
lium, carbon, aluminum, and uranium bombarded with 170Mev pro
tons have been made by Taylor, Pickavance, Cassels, and Randle (T3).
4. Neutrons from U 235 Fission. One of the most important present
day neutron sources is the fission reaction. Fission of the heaviest ele
ments is accompanied by the emission of fast neutrons. The neutron
spectrum resulting from thermal neutron fission of U 235 is closely
approximated by the expression (U2)
<W
— • = e^sinh(2£/) M (92)
dE
where E is the neutron energy in Mev. This expression, suggested by
Watt, agrees with the observed fission spectrum to within ±15 per
cent up to E = 17 Mev. 1 Equation (92) is plotted in Fig. 43.
As seen in the figure, the fission spectrum has a rather broad energy
distribution, with an average energy of ~1.5 Mev. However, this is by
no means the spectrum of neutrons normally observed either inside or
emerging from piles, since neutrons in piles suffer considerable modera
tion by elastic scattering on light nuclei or by inelastic scattering on
heavy nuclei. Nevertheless, strong sources of unmoderated fission neu
1 See T. W. Bonner, R. A. Ferrell, and M. C. Rinehart, Phys. Rev., 87, 1032
(1952); D. L. Hill, Phys. Rev., 87, 1034 (1952); B. E. Watt, Phys. Rev., 87, 1037
(1952).
Sec. 3B]
Sources and Detectors
393
trons can be obtained by irradiating a uranium target with an intense
thermal neutron beam (as from a pile). Such a source was used in the
measurements by Hughes et al. (H72) of capture cross sections for fast
neutrons, discussed in Section 2.
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.1
1 2 3 4 5 6 7 8 9 10 11 12 13
E n (Mev)
Fig. 43. Approximate shape (to within ±15%) of the neutron spectrum from the
fission of U 236 by thermal neutrons (U2).
5. Neutrons in the Cosmic Radiation. Because the neutron is un
stable, it is not expected to be one of the components of the primary
cosmic radiation. Nevertheless, neutrons constitute an appreciable frac
tion of the "hard" or "penetrating" component of cosmic rays. These
neutrons arise from nuclear disintegrations (stars) in the atmosphere
and are responsible, in turn, for a large fraction of the observed nuclear
disintegrations, especially at low altitudes (R21, S38, K22).
The observations on neutrons in the cosmic radiation are of two
general types:
(1) The detection of very fast and ultrafast neutrons by observation
of their interaction with nuclei, especially star production. Of such
experiments, the most unambiguous are those in which stars are observed
in sensitive nuclear emulsions, where the particle responsible for the
disintegration can be seen or, in the case of neutrons, deduced from the
absence of any charged particle of sufficient energy. Results of such
experiments are reported and summarized by Bernardini, Cortini, and
Manfredini (B18).
394 The Neutron [Pt. VII
(2) The detection of slow neutrons in paraffinsurrounded and
cadmiumcovered BF 3 counters (Y4, S50). Many such experiments
have been performed by many investigators; one of the most recent and
most extensive, due to Yuan (Y4), has been carried to an altitude of
~100,000 ft (pressure ~1 cm Hg). The thermal neutron intensity
shows an exponential increase (mean free path ~ 156 g/cm 2 ) up to
about 20 cm of Hg, a maximum at ~8.5 cm of Hg, and a rapid falling
off at higher altitudes.
The interpretation of experiments of the second type requires con
sideration of the processes by which the neutrons, produced mainly as
fast neutrons in nuclear disintegrations, lose energy by scattering and,
at the same time, diffuse through the atmosphere. (The general dis
cussion of such problems is reserved for Section 4.) The theory of the
slowing down of neutrons in the atmosphere has been given by Bethe,
Korff , and Placzek (B26) ; the diffusion problem is discussed by Flugge
(F35). Their calculations have recently been revised (in the light of
more accurate data on nuclear cross sections) by Davis (D5) and by
Lattimore (L6).
6. Sources of Slow Neutrons. By far the greatest number of neutron
investigations have involved the use of slow neutrons. Sources of slow
neutrons are invariably derived from intermediate or higherenergy
neutrons by allowing the neutrons to diffuse through a "moderating"
material. [The lower limit of monoenergetic neutrons from threshold
charged particle reactions, mostly (p,n), is being slowly pushed toward
the slowneutron region.]
Neutrons lose energy through inelastic collisions with medium or
heavy nuclei and through elastic collisions with light nuclei; the first
process is most effective for fast neutrons and the second for interme
diate and slow neutrons. Consider a source of fast neutrons embedded
in a large mass of material, preferably one containing light nuclei. The
neutrons diffuse through the material, losing energy as they suffer scat
terings. If the capture cross section is small compared to the scattering
cross section (as it is in the intermediate and fastneutron ranges for
most materials, especially light elements), the neutrons continue to lose
energy until their energy is comparable to the energy of thermal agita
tion of the nuclei in the moderating material. After this, a neutron is
as likely to gain as to lose energy in a scattering, and the slowing down
stops. The neutrons are then said to be in "thermal equilibrium" with
the moderating medium.
The energy distribution of neutrons in thermal equilibrium is in many
respects similar to that of the molecules of a gas at the temperature of
Sec. 3B] Sources and Detectors 395
the medium. It is closely approximated by the Maxwell distribution
function,
dn
— = Av 2 e M * 2 /M T (93)
dv
where M is the neutron mass, k is Boltzmann's constant, and T is the
absolute temperature of the moderating medium; the factor A is gen
erally a function of the position in the medium. The Maxwell distribu
tion gives as the most probable velocity
o'i^) (94)
/2kT\
\W
which is 2.2 X 10 5 cm/sec for T = 300°K; the average velocity is
2
v = —v' = 1.128?/ (95)
Strictly speaking, in an infinite homogeneous moderating medium the
thermal neutrons will have a Maxwell energy distribution only under
special circumstances, of which two are: (1) the nuclei in the moderating
medium scatter but do not absorb neutrons; (2) the neutrons are emitted
into the medium with the Maxwell energy distribution and the absorp
tion cross section obeys the 1/v law. In this case, since the rate of
neutron absorption, ~nw(v), is proportional to n, the original distribu
tion shape will maintain itself.
Most thermal neutron sources deviate somewhat from the Maxwell
distribution. If the medium strongly absorbs neutrons in the thermal
energy range, the neutrons will never attain very low velocities, and the
average velocity will be greater than that given by Eq. (95). If the
size of the slowingdown medium is not sufficiently great, the neutrons
will escape before attaining thermal equilibrium. In this situation, the
average energy will again be greater than in Eq. (95). In any event,
there is always present in the medium, at distances relatively close to
the source, a component of neutrons in the process of being slowed down.
These have an energy distribution characterized by equal fluxes (<£ = nv)
of neutrons in equal logarithmic energy intervals, i.e.,
BdE
d<j> = — (96)
where B depends both on the position in the medium and on the neutron
energy. Such neutrons (called epithermal) are observed as a tail on the
Maxwell distribution.
396 The Neutron [Pt. VII
The theory of slowing down and diffusion of neutrons will be discussed
in Section 4. Some results, useful in the planning and interpretation of
experiments involving slowneutron sources, are summarized below.
The most frequently used moderating media are water or paraffin, and
graphite. The former have the advantage of containing a large propor
tion of hydrogen, the most effective nuclei for slowing down intermediate
energy neutron; hence, for a given source strength, higher thermal
neutron densities can be obtained with smaller quantities of moderating
material than for any other slowingdown medium. On the other hand,
carbon has a considerably smaller capture cross section than hydrogen,
so that it is possible to obtain purer thermal neutron sources by making
observations at greater distances from the fastneutron source. Other
materials which have proved useful as slowingdown media include
heavy water, beryllium, and beryllium oxide (BeO); these are inter
mediate between water and graphite in slowingdown effectiveness, but
are considerably more difficult to obtain in sufficient quantity.
Since the early work of Fermi and coworkers (F17, A20) on the
moderating properties of water there have been many investigations of
slowneutron distributions in water for many fastneutron sources. The
results of one such study, due to Anderson, Koontz, and Roberts, are
shown in Fig. 44, a plot of activity X r 2 (proportional to r 2 nv) vs. dis
tance from the source. The characteristic features of such curves are
the peak at a relatively small distance and the exponential decrease at
large distances from the source.
The results of such studies are frequently summarized in terms of the
value of the second moment of the distribution,
 Ar 4 dr
* °  (97)
Ar 2 dr
where A is the measured activity of the detector used. The measure
ments of Anderson, Koontz, and Roberts (Fig. 44) give r 2 = 278 and
353 cm 2 for the indium resonance (1.44 ev) and thermal neutron detec
tor, respectively. (These are in fair agreement with the results of other
investigators.) The characteristic lengths of the exponential decrease
at large distances,
r 2 A(r)~e rl * (98)
were found to be X = 9.43 and 10.00 cm for the indium resonance and
thermal detectors, respectively.
Sec. 3B]
Sources and Detectors
397
The slowing down of neutrons in graphite has also been studied with
various sources. The results of measurements by Feld and Fermi are
shown in Fig. 45 as curves of thermal and indium resonance detector
2.0
1.6
0.2
Tr
( «
K
/
/
j
r\
/
/
1
ierrr
al n
BUtr
ons
x 1(
'
11
/
1
/
/
1/
1 1
/ 1
1 1
//
'/
(1
/
l\
Nj
1
12 16 20 24
Distance from source (cm)
28
Fig. 44. Distribution of slow neutrons from a RaaBe source in water. The ordi
nate is the activity X r 2 of a thin indium foil. The curve labeled "indium reso
nance" represents the activity of a cadmiumcovered foil; the curve labeled "thermal
neutrons" represents the activity of a bare foil minus 1.07 times the activity of the
cadmiumcovered foil. The source had a strength of 13.2 X 10 6 neutrons/sec.
The indium foils had an area of 3 X 6.4 cm 2 and a thickness of 1.69 g/cm 2 . The
cadmium covers were 0.96 g/cm 2 thick. Data due to Anderson, Koontz, and Roberts
(unpublished). The ordinate scale can be converted to a scale of thermal neutron
flux (hv) X r 1 by multiplying by 1.6 X 10 6 .
activation vs. distance from the source, for a RaaBe source (Fig. 45a)
and a Ra7Be source (Fig. 45b). Since these measurements were made
in a graphite column of nonnegligible lateral dimensions, the geometry
is not strictly comparable to that for a point source in an effectively
infinite medium (Fig. 44). However, the geometric effects can be taken
into account in interpreting such experiments (see Section 4).
398
The Neutron
RaaBe
[Pt. VII
V
180
\\
\
160
V
— Tl
erm
il
1
\
\\
120
100
\
80
\
\
\
\
•\
I
res
idiur
onar
n — 
ce
" >
•
s
120
Z (cm)
Fig. 45a. Thermal neutron flux and indium resonance neutron flux for a RaaBe
source in graphite, density 1.6 g/cm 3 , due to Feld and Fermi (unpublished). All
the data are normalized to the same value at z = 10.2 cm. The neutrons were
slowed down in a rectangular graphite column, 4 ft by 4 ft square by 7 ft high. The
source was placed on the long axis 2 ft from the bottom face of the column. The
abscissa z is distance, along the column axis, between source and indium detector.
The actual ratio of thermal to indium resonance activity at z = 10.2 cm for the
detector used was 3.72.
For a monoenergetic source in a graphite column, the distribution of
indium resonance neutrons is expected to be Gaussian:
Wm» = ae~ ( * /zo)2 (99)
The distribution of indium resonance neutrons for the RaaBe and
RaYBe sources could be expressed as a superposition of 3 and 2 Gaus
sian functions, respectively, with the proportions and ranges given in
Table 26.
Sec. 3B]
200
Sources and Detectors
RayBe
399
80
20
K
\
\
V
Th
ermj
ii
\
\
I
\
\
\
\
'
^
•
Indium
resonance
60 80
Z (cm)
100
140
Fig. 45b. Thermal neutron flux and indium resonance neutron flux for a Ra7Be
source in a graphite column (see legend of Fig. 45a), due to Feld and Fermi (unpub
lished). The actual ratio of thermal to indium resonance activity at 2 = 10.2 cm
for the detector used was 2.57. The ratio of the strength of the RaaBe source
(see Fig. 45a) to the Ra7Be source was 0.231. The corresponding ratio of thermal
neutron activities at 10.2 cm was 0.355.
The distribution of thermal neutrons is a somewhat more complicated
function of distance from the source. However, at distances greater
than ~60 cm, for the RaaBe source, and ~30 cm, for the RayBe
source, the thermal neutron density becomes exponential :
(nv) th = 0e* /x
(99a)
with the same value of X = 24.2 cm applying to both sources. It is the
difference between the exponential (Eq. 99a) and Gaussian (Eq. 99)
functions which results, in graphite, in the rapid increase of the ratio of
400 The Neutron [Pt. VII
TABLE 26
Analysis of Indium Resonance Neutbon Distributions along the Axis
op a Graphite Column fob RaaBe and Ra7Be Sources in Terms of
Gaussian Functions (Eq. 99 and Fig. 45)
RaaBe Ra7Be
«» (%) «o (cm) oci (%) z (cm)
15.0 22.8 38.7 22
69.3 36.9 61.3 29
15.7 57.1
meanf 39.2 meant 26.5
t The mean is computed as the square root of the weighted mean square.
thermal to epithermal neutron flux with increasing distance from the
source.
While slowneutron sources of the types described above have been
and continue to be extensively used, the most potent available slow
neutron sources are those associated with chain reactors (piles). The
spectrum of neutrons inside a slowneutron reactor contains an appre
ciable component of fast, intermediate, and epithermal neutrons. 1 How
ever, by use of a "thermal column," it is possible to obtain high fluxes
of practically pure thermal neutrons. The usual thermal column is
simply a large block of graphite placed directly adjacent to the pile.
Neutrons which enter the column are slowed down as they diffuse to the
outside, the ratio of epithermal to thermal neutrons falling off rapidly
with the distance into the column. Figure 46 shows the activity of bare
and cadmiumcovered indium foils at various positions in the thermal
column adjacent to the Argonne heavywatermoderated pile.
Thermal neutron fluxes, available from various sources, are summar
ized in Table 27.
For many purposes, the thermal neutrons diffusing in a mass of
moderating material can be utilized directly, in situ. Thus, for example,
in irradiating a sample for the purpose of observing the resulting radio
activity, the maximum slowneutron flux and consequently the maxi
mum activation are attained when the sample is embedded in the
moderator as close as possible to the source.
For some investigations, on the other hand, it is desirable to have a
beam of thermal neutrons, e.g., for the measurement of cross sections by
a transmission experiment. One method of obtaining a beam of thermal
neutrons is to collimate (with a cadmium or boron slit system) a portion
of the neutrons which are diffusing out through the bounding surface of
1 For the spectrum of neutrons leaving a pile, see Fig. 54.
Sec. 3B]
Sources and Detectors
401
10,000
1000
0.001
20
71 cm>j
from face
of tank
100 120 140 160 180 200 220 2
Cm from face
260 280
End of
graphite
Fig. 46. Activities of bare and cadmiumcovered indium foils in the thermal column
of the Argonne heavy water pile. The thermal column is of graphite, 5 ft square and
240 cm long. Its front face is separated from the reactor by 71 cm of graphite. The
average thermal neutron flux (nv) can be obtained by multiplying the ordinates
corresponding to the bare indium foils by 0.10. The number of neutrons per cubic
centimeter passing through the indium resonance energy per second (slowingdown
density) can be obtained by multiplying the ordinates corresponding to the cadmium
covered indium foils by 0.0019. Measurements due to Seren (unpublished).
402
The Neutron
[Pt. VII
TABLE 27
Some Available Thermal Neutbon Fluxes
(a) Laboratory Sources
Strength
Thermal nv
Source
(n/sec)
Position
(n/cm 2 • sec)
1 curie of RaaBe in
13.2 X 10«
4 cm from the source
8.5 X 10 4
infinite H 2
10 cm from the source
2.9 X 10 4
20 cm from the source
0.31 X 10 4
50 cm from the source
0.0025 X 10 4
1 curie of RaaBe in
14 X 10 6
at source
0.7 X 10 4
graphite column,
10 cm from the source,
0.65 X 10 4
4 ft x 4 ft x 7 ft;
on the axis
source 2 ft from
50 cm from the source,
0.25 X 10 4
base, on axis
on the axis
100 cm from the source,
0.04 X 10 4
on the axis
2 curies of RayBe
3.2 X 10 6
at source
0.3 X 10 4
(608 grams Be) in
10 cm from the source,
0.2 X 10 4
graphite column,
on the axis
as above
50 cm from the source,
on the axis
0.07 X 10 4
100 cm from the source,
0.008 X 10 4
on the axis
(b) The Center of a Nuclear Reactor '
Pile Type
Location
Power
Level
(kw)
Flux
(n/cm 2 •
sec)
Heterogeneous
Graphite — U
Oak Ridge, Tenn., U.S.A. 2
Harwell, England (GLEEP) 3
(BEPO) 4
~10 3
4X 10 3
2 X 10 12
10 8
10 12
Brookhaven, L.I., U.S.A. 5
2.8 X 10 4
4 X 10 12
Homogeneous
H 2 0— U 236
Los Alamos, N. Mex., U.S.A.
(HYPO) «
6
3 X 10 u
Heterogeneous
D 2 0— IT
Argonne, 111., U.S.A. (CP3) '
Chalk River, Canada (NRX) '
300
>10 4
10 12
5.8 X 10 13
Chatillon, France (Zoe) 8
—10
3 X 10 10
Oslo, Norway 9
~100
3 X 10 u
Sec. 3B] Sources and Detectors 403
TABLE 27 (Continued)
Some Available Thermal Neutron Fluxes
(c) Pile Thermal Columns
Flux
Pile Position (n/cm 2 • sec)
Argonne (CP3) 10 inner face 3 X 10 s
Chalk River (NRX) " 1 meter from inner face 10 9
Chatillon (Zoe) 8 outer face 2 X 10 4
(d) Holes in the Shield
Pile
Flux (n/cm 2 ■ sec)
Oak Ridge 10
10 7
Chalk River (NRX) "
4X 10 7
Chatillon (Zoe) 8
10 6
1 D. J. Hughes, Nucleonics, 6 [2], 5(1950); H. S. Isbin, Nucleonics, 10 [3], 10
(1952).
2 A. H. Snell, Nucleonics, 8 [3], 3 (1951).
3 F. C. W. Colmer and D. J. Littler, Nucleonics, 8 [1], 3 (1951).
* Nucleonics, 8 [6], 36 (1951).
6 Physics Today, Jan. 1951, p. 6.
6 Nucleonics, 7 [6], 2 (1950).
7 Nucleonics, 6 [3], 77 (1950); F. W. Gilbert, Nucleonics, 10 [1], 6 (1952).
8 L. Kowarski, Helv. Phys. Acta, 23, Supp. 3, 70 (1950).
9 Bulletin of the Atomic Scientists, 7, 380 (1951).
10 Preliminary reports (unpublished) for NRC Subcommittee on Neutron
Standards, by Bernstein and others.
11 E. Almquist, Can. J. Research, A28, 433 (1950).
12 D. G. Hurst, A. J. Pressesky, and P. R. Tunnicliffe, Rev. Sci. Instr., 21,
705 (1950).
the moderating medium. However, the beam intensities, which can be
attained by this means (especially with easily available sources), are
very low, since relatively large masses of moderating material are re
quired to thermalize the source neutrons, and the neutron densities, at
the surface of the moderator, are consequently low; in addition, the
emerging neutrons have a relatively broad angular spread, so that most
of them are absorbed by the collimating slits.
The common method of attaining strong thermal neutron beams is
to use a "howitzer," which is a relatively large mass of moderator (usu
ally paraffin) with a hole reaching down to a region of high thermal neu
tron flux, close to the source. The hole is usually lined with cadmium
404 The Neutron [Pt. VII
or boron or both, thereby providing the necessary collimation for the
beam. By proper design, it is thus possible to obtain relatively strong,
wellcollimated thermal neutron beams, even when only radioactive
(a,n) sources are available (J 13).
C. Neutron Detectors
1. Induced Radioactivity. Neutrons are capable of inducing a variety
of nuclear reactions, many of which result in radioactive product nuclei.
Thus, neutrons can frequently be detected by the resulting radioactivity
of the exposed substance.
Radioactivity is detectable in many ways: Geiger counters for beta
rays of energy sufficient to penetrate the counter walls or, with lesser
efficiency, for gammarays; electrometers or ionization chambers for
alpha, beta, or gammaradiation; scintillation counters, electron multi
pliers, etc. The feasibility of detecting the radioactivity resulting from
neutron exposure depends on the lifetime of the induced radioactivity;
it cannot be appreciably shorter than the time which must (for experi
mental reasons) elapse between exposure to neutrons and measurement
of the resulting radioactivity; on the other hand, the lifetime must not
be so long that the radioactive decay rate is negligible. These con
siderations normally have the effect of limiting the lifetimes of possible
radioactivities to between ~10 — x sec and ~40 4 years.
Consider a detector which absorbs neutrons at the rate of Rq per
second, and whose consequent radioactivity decays with a mean life of
r sec. The number of radioactive nuclei, N*, is governed by the differ
ential equation
dN* N*
— = R (100)
dt r
The activity of the detector (disintegrations per second), t sec after a
(constant) exposure of duration t e , is
R(t,t e ) = 7?o(l  e~ l ' /T )e t/T (100a)
The observed activity (usually counting rate),
R'(t,Q = &R(t,t e ) (101)
contains the factor S, the efficiency of the detecting system; S depends
on the properties of the radiations involved, on the geometry of the
absorber and detector, and on the efficiency of the radiation detector.
The activity which would be observed immediately after an infinite
irradiation,
R'(t,t e )
R ' = #'(£ = 0, t e = oo) = __ _ = S R Q (ioia)
(1 — e ( «/')e _ '/ r
Sec. 3C] Sources and Detectors 405
is usually referred to as the "saturated activity." Its value is inde
pendent of the schedule of neutron exposure and radioactivity measure
ment.
Ro' is usually measured by integrating the activity over a finite energy
interval, say from t = t x to t  t 2 . The integrated activity is
C(h,ta,Q = f t 'R'&Q dt = i\R'(ti,U) ~ R'(t 2 ,te)] (102)
and the saturated activity is
p , C(h,k,t e )
Ro — (102a)
r(l  e ( «A)(e ( iA  e'sA)
The saturated activity is a measure of the neutron flux impinging on
the absorber. If S is known, the radioactivity measurements yield R ,
which can be directly interpreted in terms of the neutron flux <j> = nv.
We consider two types of measurement:
(1) The absorber is embedded in an isotropic neutron flux whose
spectrum is given by <j>(E), the density of neutrons per unit energy
interval, at the energy E, multiplied by the neutron velocity correspond
ing to E. If, furthermore, the absorber is weak, 1 the rate of neutron
absorption is
R = Nvf<j>(E)<r a (E) dE (103)
where N is the density of absorbing nuclei and V the volume of absorb
ing material. (NV is the total number of absorbing nuclei.)
In general, both cf>(E) and a(E) are relatively complicated functions
.of E, and a measurement of R serves only to determine the integral.
However, there are some interesting special cases: Suppose that the
neutron density has a Maxwell distribution
*(») = vM(v) = Av 3 e  (v/v ' )2 (93a)
and the cross section follows the 1/v law,
<r a (v) = (94a)
v
Then
R Q = NVc a 'v'A f v 2 e~ <»A') 2 dv
NVAW)*tr a '7rX
4
1 By which we mean that its presence does not disturb the neutron density,
(103a)
406 The Neutron [Pt. VII
The total flux is x
$ = I M(v) dv =
Jo
A{v')
iM
whence
2
(93b)
NV$a a 'ir 1A NV$c a '
R = = ( 10 3b)
2 1.128
Thus, for a Maxwell neutron distribution and al/» detector, the mean
detector cross section is
_ er a V IT 7 * , <f a
fa = (95a)
,v 2 1.128
where a a ' is the absorption cross section corresponding to the neutron
energy E' = kT.
Suppose, on the other hand, that the absorber has a single sharp
resonance at the energy E r . (The effect of thermal neutron absorption
can be eliminated by surrounding the detector with cadmium.) The
cross section can be represented by a deltafunction:
c a {E) = P 8(E,E r ) (104)
where
p = j<r a (E)dE = <r r
(104a)
for a resonance, of peak cross section <r and width r, which follows the
BreitWigner formula. Provided that <t>(E) does not vary appreciably
over the resonance,
Ro = NV P <t>{E r ) (103c)
When, as is frequently the case in the resonance region,
4>{E) =  (96)
Hi
Eq. (103c) gives
NVpB
Ro = —=— (103c')
E r
If more than one resonance is involved,
Ro = NVBYi ~ (103c")
i Eri
The expression
E T = J v ~V~ (104b >
is called the "resonance integral."
Sec. 3C] Sources and Detectors 407
Usually, the insertion of an absorber into the neutron flux causes a
perturbation of the flux measured, so that the activity of the detector is
not an exact measure of the true flux in the absence of the absorber.
This perturbation arises from two causes: a reduction of the neutron
density inside the detector which is due to the absorption of neutrons
by the outside layers; and a depression of the flux directly outside the
detector due to the finite neutron absorption. Such effects have been
considered by Bothe (B53), who has derived a correction factor /for the
reduction of the specific activity of a sphere of radius R,
1 a
 = 1 +
/ 2
3RL
~2\^R+~L)~ .
for R » \ ir (105a)
1 0.34aK
 = 1 + — for R « X tr (105b)
J A«r
In the above expressions, L is the diffusion length (see Section 4) of the
neutrons in the medium outside the detector; \ tr is their transport mean
free path; a is the average probability that a neutron will be absorbed
in a single traversal of the detector. For an isotropic flux
■ « = 1 _ e » T {l  pT) + n 2 T 2 Ei{».T) (105')
where ft = N<r a is the absorption coefficient, in cm 1 , of the detector
and T is the average absorber thickness. (T = R for a spherical ab
sorber.) Ei{ — x) is the exponential integral.
Tittle (T12) has shown experimentally that the above expressions
also apply without modification to an absorber in the form of a disk of
radius R and thickness T.
(2) The second situation of interest involves neutron beams. Con
sider a neutron detector of area (perpendicular to the beam direction)
A and thickness T. The absorption rate is
Ro = aU(E)(1  e" r )
dE
(103')
An absorber for which fiT = NTc a (E) » 1 is said to be "black" for
neutrons of energy E. Since
ft (black) = A J 0(2?) dE (103 'a)
such an absorber gives a direct measure of the total neutron flux. At
408 The Neutron [Pt. VII
the other extreme, a "transparent" detector is one for which fiT <JC 1, in
which case
R (transparent) = ATN f<t>(E)* a (E) dE (103'b)
Owing to the large variation with energy of slowneutron cross sec
tions, most absorbers are transparent at some energies, black at some,
and translucent at others.
2. The SzilardChalmers Reaction. In many nuclear reactions, e.g.,
(n,p), (n,a), (ft,/), the product nuclei differ in their chemical properties
from the target nuclei, and can therefore be separated from the target
with relative ease. For such detectors the radioactivity from large
masses of absorber can be reduced to moderately small samples for
counting, and the absolute detection efficiency can be made relatively
high. However, many of the important neutron reactions — (n,y), {n,n'),
(n,2n) — lead to isotopes of the target element. For such detectors the
specific activity of the absorber may be too small to permit accurate
measurement of relatively weak neutron fluxes.
To overcome these difficulties Szilard and Chalmers (S62) in 1934
devised a technique for separating radioactive nuclei, induced by the
(n,y) reaction, from their isotopic environment. This technique takes
advantage of the fact that the emission of gammaradiation after slow
neutron capture causes a recoil of the product nucleus; the recoil energy
is usually sufficient to disrupt the molecular bond and, thereby, to
change the chemical state of the product nucleus as compared to a nor
mal nucleus in the medium. The radioactive nuclei can then, in favor
able circumstances, be separated chemically from the normal nuclei,
with efficiencies as high as 100 percent.
For the emission of a single gammaray of energy E y (in Mev) from a
nucleus of mass number A, the nuclear recoil energy E (in ev) is
536E y 2 , N
E = (106)
For typical values, E 7 = 7.5 Mev and A = 100, the recoil energy is E
= 300 ev, which is considerably in excess of that usually necessary to
disrupt chemical bonds, say ~5 ev. Even if two or more gammarays
are emitted after slowneutron capture, all but a few percent of the re
coiling nuclei will usually have E > 5 ev.
In their original experiment Szilard and Chalmers irradiated ethyl
iodide with neutrons from a RaaBe source (most of the absorptions
Sec. 3C] Sources and Detectors 409
are due to slow neutrons) and extracted the radioactive I 128 (25min
halflife) by shaking with an aqueous reducing solution containing a
trace of free iodine as a carrier. Similar techniques have been applied
to other halogens, and methods have been worked out for separating
the product nuclei, due to the (n,y) reaction, of many other elements.
Specific SzilardChalmers processes for different substances are de
scribed and discussed in excellent summaries (with references) of the
technique by McKay (M14), by Broda (B68), and by Barnes, Burgus,
and Miskel (W2).
The SzilardChalmers reaction has frequently been used to obtain
radioactive samples of high specific activity, especially when only rela
tively weak neutron sources were available. Thus, von Halban, Kowar
ski, and Magat (H6) employed a solution of bromine to measure the
intensity of neutrons in the cosmic radiation. The strength of relatively
weak fastneutron sources is frequently measured by stopping the
neutrons in a water solution of potassium permanganate, and removing
the radioactive Mn 56 (2.6h halflife) as Mn0 2 by filtration through
fine paper or sintered glass (E6) ; the efficiency of this SzilardChalmers
reaction is greatest for a neutral or slightly acid solution. Broda and
Rieder (B67) have shown that the same reaction can be used to detect
fast neutrons.
3. Ionization Chambers and Proportional Counters. Many neutron
reactions involve the prompt emission of a charged particle. Such reac
tions can be detected, through the ionization caused by the emitted
particle, in an ionization chamber or a proportional counter. In addi
tion to whatever other distinctive properties such detectors may possess,
they have the advantage over induced radioactivity that they can be
used to observe neutrons in situ during the irradiation, and will reflect
shorttime fluctuations in the neutron density; a radioactive detector,
on the other hand, is inherently a timeintegrating device.
Owing to the relatively short range of the heavy charged particles
resulting from neutron reactions, the target nuclei must, in general,
be inside the chamber. They can be introduced into the chamber
either as a constituent of the gas with which the chamber is filled (and
in which the detected ionization takes place) or as a coating on the
inside walls.
In ionization chambers the ionization of the gas is measured (without
amplification in the chamber) either as a current or, in the case of pulsed
ionization chambers, as individual pulses associated with single nuclear
disintegrations. Proportional counters differ only in that the ionization
is amplified in the gas of the chamber. In both cases, the observed
charge or pulse height is proportional to the energy dissipated in the
410 The Neutron [Pt. VII
chamber (although the constant of proportionality may vary somewhat
with the position in the chamber at which the event occurs).
The construction and use of ionization chambers and proportional
counters has been discussed in Part I of Volume I. The following is
intended as a brief summary of the main considerations involved in the
use of such devices to measure neutron fluxes: When a neutrondetecting
ionization chamber or proportional counter is placed in a neutron flux
<j>(E), the observed intensity, / (ionization rate or counting rate), is
determined by the same factors as the saturated activity R ' of a radio
active detector (Section 3C1) with the important difference that the de
tection efficiency is, in general, a function of the neutron energy. Thus,
we have (neglecting the variation of efficiency with position in the cham
ber) : for a chamber in an isotropic neutron flux,
I = NV f&(E)^(E)a(E)f(E) dE (103")
where the symbols are as previously defined and f(E) is given by Eqs.
(105a), (105b), (105'); for a chamber in a neutron beam,
I = A £&(E)<}>(E)(1  e* r ) dE (103'")
The efficiency function, &(E), depends on the particular reaction used,
on the chamber geometry, on the method of introduction of the reacting
nuclei, and on the particular detection method used. The following dis
cussion applies only to neutron reactions in which all the energy is car
ried away by charged particles. Reactions involving the emission of
neutrons as well as charged particles (of which the most important is
neutron scattering, especially by protons) will be taken up in a separate
section. We consider a number of the most widely used detection
schemes :
(la) / = ionization current; the reacting nuclei are in the chamber
gas:
8(E) = XX (E + Q) X G{E) (107)
where E is the neutron energy and Q is the reaction Q value. (Note : For
Q < 0, S = for E < [(A + 1)/A]Q.) G(E) is a geometric factor
which depends on the chamber shape, on the gas pressure, and on the
rangeenergy relationship for the emitted charged particles in the cham
ber gas. It represents the average (over the chamber) fraction of the re
action energy which is dissipated in the chamber gas; G(E) — > 1 as the
range of the emitted particles becomes small compared to the chamber
dimensions. The calibration constant K can be determined by measur
Sec. 3C] Sources and Detectors 411
ing the ionization produced by a known flux of known energy; alterna
tively, K can be determined by observing the ionization resulting from
the insertion into the chamber of a calibrated source of polonium alpha
particles.
(lb) / = ionization current; the reacting nuclei are introduced as a
coating on the chamber walls. If the coating is thin compared to the
range of the emitted charged particles, the situation is similar to that
for a gasfilled chamber, except that only ~ \ the reaction energy is dis
sipated in the chamber gas. As the coating thickness is increased, the
emerging charged particles will lose energy in traversing the coating.
For a thick coating (T > range of the emitted charged particles) only
those reactions taking place within a layer of thickness equal to the
range of the emitted charged particles can be effective in producing
ionization in the chamber gas. The efficiency function of an ionization
chamber with a given coating can, in general, be computed. In addition
to its dependence on the specific properties of the reaction, the effi
ciency may also depend on the angular distribution of the neutron flux
being detected.
The form of &(E) is quite different for the counting of individual
pulses. In general, the associated electronic equipment is designed so
that only pulses of magnitude greater than a predetermined value are
counted. (This type of discrimination makes it possible to use pulse
counters even when the neutron flux is accompanied by a considerable
background of gammaradiation.) A pulse detector in which all pulses
greater than a given size are counted is called an "integral" detector.
It is also possible to arrange the electronics so that only those pulses
are counted whose sizes lie between two definite (usually variable)
limits; such arrangements are called "differential" detectors.
(2) I = counting rate; integral detection. If the reacting nuclei are
in the chamber gas, &(E) is, as a first approximation, given by a step
function, having the value zero for neutron energies less than E t , and
one for E > E t ; E t is the neutron energy for which the ionization of the
reaction products is just sufficient to produce a pulse of the required
size. However, in a chamber of finite size the step function must be
modified by a geometric factor G(E), to take into account the possibility
that some products of an otherwise detectable reaction may dissipate
part of their energy in the chamber walls and give rise to pulses smaller
than the detectable minimum.
When the target nuclei are coated on the chamber walls the same
considerations apply as in scheme (lb).
(3) / = counting rate; differential detection, with the reactions occur
ring in the gas of the chamber. Ideally (for a chamber of infinite dimen
412 The Neutron [Pt. VII
sions) &(E) is given by the difference between two step functions with
thresholds E t \ and E t 2 In practice, the effect of the finite size of the
chamber is to make S smaller than 1 between these limits and to add to
&(E) a tail extending beyond E t2 .
When the reacting nuclei are introduced as a wall coating, differential
detection is useful only if the coating thickness is small compared to the
range of the reaction products.
(a) Detection by the B 10 (n,a) Reaction. Because of its large cross sec
tion, the B 10 (w,a) reaction is extensively used in slowneutron detectors.
The reaction is highly exoergic, Q = 2.78 Mev. However, most slow
neutron captures lead to the 0.48Mev excited state of Li 7 (which decays
to the ground state by gammaray emission); the ionization energy re
leased in such captures is 2.30 Mev. Bichsel, Halg, Huber, and Stebler
(B29) and Petree, Johnson, and Miller (P14) have measured the ratio of
captures leading to the ground and to the 0.48Mev excited states for
neutron energies from thermal to 3.9 Mev. The ratio follows a smooth
curve, from ~0.07 for thermal neutrons to a maximum of somewhat
greater than 2 at 1.9 Mev, and then falls to ~1 above 2.5 Mev.
The (n,a) cross section of normal boron element follows the 1/v law
up to energies of at least 1 kev, with <r'(y' = 2.2 X 10 5 cm/sec) = 710
barns. Its behavior in the intermediate range is not so well established,
but its general features have been determined; there is a resonance at
1.9 Mev (G20, A2, P14).
The isotope B 10 has a natural abundance of 18.83 percent. Its iso
topic (n,a) cross section is greater than that for natural boron by the
factor 5.31 ; i.e., </(B 10 ) = 3770 barns. 1 The availability of B 10 enriched
boron thus makes possible the construction of neutron detectors with
high efficiency over a wide range of neutron energies.
Boroncontaining ionization chambers and proportional counters have
been operated under a variety of conditions, as currentmeasuring de
vices and disintegration counters, with the boron introduced as a gas
or as a wall coating. Because of the large Q value, 8(E) is essentially
energy independent in the slow and intermediateneutron energy
ranges. Furthermore, the large Q value makes it easy to discriminate
between pulses resulting from neutron capture and background pulses
arising from relatively strong gammaray intensities; hence, boron
containing chambers are normally operated as integral pulse counters.
BF 3 is usually used as the chamberfilling gas. It has relatively good
ionization chamber and proportional counter characteristics, provided
that the gas is free of impurities. Techniques for the construction, fill—
1 See the footnote to Table 11, p. 339.
Sec. 3C] Sources and Detectors 413
ing, and operation of BF 3 counters have been discussed by a number of
authors (S15, B31, F39, B56).
For a chamber containing BF 3 at one atmosphere pressure, the slow
neutron absorption coefficient is
M (normal BF 3 ) = 0.0191 (0.025/E)' A cm 1
(i (B 10 F 3 ) = 0.101 (0.025/^)* cm 1
(E is in ev.) Thus, a typical counter (of average thickness ~25 cm)
would vary in effectiveness from black to transparent over the ther
mal neutron range. However, most BF 3 counters — even B 10 F 3 — are
transparent for epithermal neutrons. For a slowneutroninduced dis
integration in such a chamber, the range of (93% of) the alphaparticles
is ~0.8 cm. (The Li 7 fragment, which carries off f{ i of the energy, has
a much smaller range.) Thus, for chambers of reasonable dimensions,
the wall effect correction G(E) is relatively small, and &(E) « 1.
Chambers in which the boron is introduced as a wall coating are less
extensively used for slowneutron detection. The thickness of the coat
ing must be kept very small, both because of the small range of the reac
tion products and because of the strong neutron absorption of boron (if
the neutrons must penetrate the coating from the outside). Hence, such
chambers are usually quite transparent to neutrons. It is possible to
decrease the transparency, without decreasing the efficiency, by design
ing the chamber to contain a multiplicity of thin boron layers (L24).
(b) The Long Counter. The sensitivity of BF 3 counters decreases
rapidly with neutron energy, becoming prohibitively small, for most
counters, in the intermediateenergy range. In order to increase the
sensitivity to intermediate and fast neutrons, experimenters have fre
quently immersed the counter in a moderating medium, usually water
or paraffin. However, because of the strong dependence of the sensi
tivity of such arrangements on the source energy and on the geometry,
the results of this method of neutron flux measurement are usually very
difficult to interpret, especially when the source has a heterogeneous
energy distribution.
Hanson and McKibben (H32) have devised an arrangement which is
uniformly sensitive to neutrons from ~10 kev to ~3 Mev. This ar
rangement, the "long counter," consists of a cylindrical BF 3 counter
(diameter ~J in., length ~10 in.) placed along the axis of a paraffin
cylinder ~8 in. in diameter. Two of their successful long counter de
signs are shown in Fig. 47.
The long counter is used in observations on neutron beams which
enter the circular face (righthand side in Fig. 47a) of the arrangement
preamplifier
Aluminum tube
Long Counter, 8"0.D.
Case of 0.050" sheet iron.
Removable aluminum
cylinder containing
BF, tube
High voltage
3'^) J*
Shielded Long Counter
(a)
2 3 4
Neutron energy (Mev)
(b)
Fig. 47. (a) Two long counters, designed by Hanson and McKibben (H32), with
uniform sensitivity (to within ~10 percent) from 10 kev to 3 Mev. The second, or
shielded, counter is designed to operate in a region of relatively high scattered neu
tron background, (b) Sensitivity vs. neutron energy of the shielded long counter
with and without holes in the front face of the paraffin:
414
Sec. 3C] Sources and Detectors 415
and diffuse parallel to the axis. The second arrangement shown is de
signed for situations in which there is an appreciable background of
scattered neutrons impinging on the paraffin from all directions. The
uniform energy sensitivity is based on the approximate cancellation of
two effects: (1) lowenergy neutrons do not penetrate far into the paraf
fin, and have an appreciable probability of being turned around and
escaping out of the front face; (2) highenergy neutrons penetrate much
farther before being slowed down, but have an appreciable probability
of escaping through the sides. The success of the arrangements of Han
son and McKibben is based on a proper choice of the diameter of the
paraffin cylinder. In the second design, holes can be bored into the front
face to decrease the loss of lowenergy neutrons (Fig. 47b).
For the shielded counter shown in Fig. 47a, the sensitivity does not
vary by more than ~10 percent between 10 kev and 3 Mev, as shown in
Fig. 47b. At lower energies the sensitivity decreases to ~70 percent for
thermal neutrons; at higher energies the sensitivity again falls off (it is
~8595 percent for RaaBe neutrons). The absolute sensitivity of
the counters is such that they give ~1 count per 10 s neutrons emitted
isotropically from a source 1 meter from the face.
(c) Fission Chambers. Fission of the heaviest nuclei by neutrons
provides an ideal source of ionizing particles for pulse detectors. Since
the fission fragment energies are ~100 Mev and their ranges are very
small, the pulse due to a fission fragment is easy to distinguish from the
background due to other ionizing radiation (e.g., pulses due to alpha
particles, spontaneously emitted from most fissionable nuclei). Thus,
fission chambers are almost always operated as integral pulsecounting
ionization chambers.
The fissionable nuclei may be introduced into the chamber as a gas
(e.g., UF 6 ) or as a wall coating. Owing to the general unavailability of
gases containing the heaviest elements or the inconvenience in handling
these gases, or both, the latter method is usually used; uranium, for ex
ample, is easily obtainable in a solid oxide or nitrate form. In either
event, the efficiency function, &(E), is essentially independent of neutron
energy, since the Q value of the (n,f) reaction is so great. (We neglect
the possibility that the coating may be so thick as to prevent the neu
trons from penetrating into the chamber.)
Fission chambers containing thermally fissionable nuclei (U 233 , TJ 235 ,
Pu 239 ) are efficient thermal neutron detectors. Chambers containing
natural uranium, or uranium enriched in the U 238 isotope, can be used
as fastneutron detectors, with an effective threshold of ~1.5 Mev;
thorium, protactinium, and neptunium can also be used in the same
fashion (see Section 2C4).
416 The Neutron [Pt. VII
Bismuth undergoes fission by ultrafast neutrons; it has a threshold of
~50 Mev and a cross section, for ~84 Mev neutrons, of ~0.05 barn
(K7, W26). Ionization chambers coated with bismuth have been used
as specific ultrafastneutron detectors (W26).
(d) Proton Recoil Detectors. Perhaps the most common method of
detecting and measuring fastneutron fluxes involves the observation of
proton recoils from neutronproton scattering. Use of this reaction has
the advantage that the cross section is comparatively large (14 barns
at E = 0.1 Mev; 4 barns at E = 1 Mev; 1 barn at E = 10 Mev; see
Fig. 2). The main disadvantage of the proton recoil method is that for
a given incident neutron energy, E n , the proton recoil energies, E, are
uniformly distributed between and E n ; the recoil energy as a function
of the angle 6 in the laboratory coordinate system (0 is confined to the
interval 090°) is
E = E n cos 2 6 (108)
Let us assume a monoenergetic neutron flux <j>(E n ) incident on a trans
parent chamber containing NV atoms of hydrogen. Neglecting, for the
moment, the effects of the finite dimensions of the chamber, the number
of pulses corresponding to the proton energy E is
NV<j>(E n )<r(E n ) cLE
P(E n ,E) dE = VK I ~ (109)
E n
for E <E n ; P(E n ,E) = for E > E n .
The chamber may be operated as an integral counter, in which all
pulses are counted whose energy exceeds the bias energy E t . In this
case, the counting rate is
E " IV(tf n )
I(E n ,E t ) = f P(E n ,E) dE = NV4>(E n )
JBt
(E n  E t ) (110)
Proton pulses of energy less than ~0.1 Mev are usually too small to
be distinguished from the background. In the region 0.110 Mev, the
neutronproton scattering cross section follows, to a good approximation,
a l/v law; i.e., <r(E n ) ^ (E t /E n ) y2 <r(E t ). In this energy range
I(E n ,E t )^NV<j>(E n )a{E t ) (—J
= NV<j>(E n )<r(E
E n
l~E t ~ l
«Kt)
(110a)
The sensitivity of such an integral pulse counter is plotted as the solid
curve in Fig. 48. S(E n /E t ) has the property, in this energy range, of
Sec. 3C]
Sources and Detectors
417
being relatively independent of the neutron energy; its value is within
30 percent of the maximum for 1.52^ < E n < U.5E t .
For neutrons of E n > 10 Mev, the neutronproton cross section can
be approximated by <r(E n ) = {E t /E n )c{E t ), whence
«®®'[®'
(110a')
This sensitivity function is plotted as the broken curve in Fig. 48. In
1.0
f
I 0.!
I 0.6
8
0.4
7
/
1 i
f\
II
if
\
\
1/
41 H
S
\
1
>
If
(J
v
X
1
•».
1
li
^».

>■
\\
—

— ~r=r 0.2
! 10
16
18
Tig. 48. The sensitivity function S(E n /E t ) of an integral pulsecounting hydrogen
filled chamber (neglecting wall effects) vs. x = E n /E t . The solid curve is for <r(E n )
= (Ei/E n ) ' A cr(Et), a good approximation to the neutronproton scattering cross
section for the range 0.1 Mev < E n < 10 Mev. The broken curve is for a{E n )
= (E t /En)<r(Et), which applies for E n > 10 Mev. The curves are normalized to a
value of 1 at the maximum.
this energy range the integral counting rate is much more strongly de
pendent on E„, However, for neutron energies above ~20 Mev, the
angular distribution in neutronproton scattering is no longer spherically
symmetrical in the centerofmass coordinate system, and the formulas
developed above must be correspondingly modified.
Hydrogenfilled chambers are frequently operated as ionization cham
bers. For such operation the chamber sensitivity to neutrons of energy
E n is
I'{E n ) = I 1
J n
P(E n ,E)E dE =
NV4>{E n )*(E n )E n
(111)
for neutrons of energy up to ~20 Mev, and neglecting wall effects. How
ever, ionization chambers are particularly sensitive to backgrounds of
all kinds. Since most neutron fluxes are accompanied by an appreciable
418 The Neutron [Pt. VII
gammaray intensity, the background presents a serious problem. One
device for eliminating such background effects is to employ two chambers
of identical geometry, one filled with the hydrogencontaining gas and
the second filled with argon. The external electronic amplifications are
adjusted in a pure gammaray beam so that both chambers give the
same current. The difference in the currents, when the chambers are
placed in a neutron flux, can be attributed to the proton recoils from
neutron scattering.
The effect of the finite dimensions of the chamber is to decrease the
sensitivity by a factor which depends on the neutron energy and which
can be computed for most chambers. The geometric factor, G(E n ), is a
monotonically decreasing function of E n . Unfortunately, the recoil
proton ranges are in general comparable to the chamber dimensions, so
that G(E n ) usually represents a sizable correction. For a given cham
ber geometry the correction becomes less important as the gas pressure
is increased. Thus, the development of techniques for the operation of
ionization chambers and counters at high pressure (W35) is of consider
able importance for the future of fastneutron spectroscopy. Alterna
tively, since organic phosphors (solid and liquid) have a high hydrogen
content, it is possible to employ such materials for fastneutron counting
by observing the scintillations resulting from the recoil protons (J3,
06) ; x the recoil proton range is usually small compared to the dimen
sions of the phosphor. Unfortunately, the pulses from such counters do
not appear to be strictly proportional to the recoil proton energy.
The above discussion applies to monoenergetic neutron sources. If
the neutron source is heteroenergetic, the pulse height distribution is a
superposition of pulses due to all the neutrons :
r°° r x <t>(E n )a(E n ) dE n
P(E) dE = I P{E n ,E) dE n dE = NV dE I — — — — (109')
Je J e E n
The function <j>(E n )a(E n )/E n can be obtained by differentiation of the
curve of P(E) vs. E. However, in order to determine <t>(E n ) vs. E n to a
reasonable accuracy, the curve of P(E) vs. E must be determined to a
much greater — frequently unattainable — accuracy. The use of integral
counting to obtain 4>{E n ) vs.'E n is even less satisfactory in that it re
quires exceptionally high precision.
When the incident neutrons are in a beam (unique direction of motion)
it is possible to choose, for counting, only those proton recoils whose
directions are the same as that of the incident neutrons. In such a de
1 An ingenious application of this principle has been devised by W. F. Hornyak,
Rev. Sci. Instr., 23, 264 (1952).
Sec. 3C] Sources and Detectors 419
vice it is necessary to collimate the protons. However, the collimation
need not be too sharp; a proton recoil at, say, 20° from the incident
neutron direction has an energy E = 0.88^„ (Eq. 108). Assuming a
collimating system which accepts protons within the relatively small
angle 6 , and neglecting the small variation of pulse heights within the
cone of acceptance,
P(E) = NV<KE)*(E)f(fi ) (109")
where
AE 6 2
f(!h) = — = 1  cos 2 O = — (108a)
(d may be a function of the proton recoil energy E.) The smaller the
angle do, the smaller is the value of f(ff), but the sharper the energy reso
lution of the device. Thus, in common with almost all problems in
spectroscopy — neutrons or otherwise — the practically attainable reso
lution is primarily determined by the strength of the source.
Many arrangements can be conceived for collimating the observed
protons. One possibility is to detect the protons in a long cylindrical
chamber, the incident neutrons traveling along the chamber axis, and
to divide the chamber into two or more independent regions, requiring
coincidences between adjoining regions. In some cases, the last section
is operated in anticoincidence with the preceding sections, thereby de
fining the range of the detected protons. Another possibility is to
separate the sections by barriers, with holes in them for the proton
collimation. The hydrogen may be introduced into the chamber as a
gas or, more usually, as a thin radiator at the incident face of the cylinder.
A thin radiator is, by definition, one in which a proton, originating in
the back of the radiator, loses only a small fraction of its energy before
emerging into the counter. The figures in Table 28 give the approxi
TABLE 28
Some Typical Proton Recoil Ranges
Range
Range
E p
Range
(mg/cm 2
(mm
Mev)
(cm std. air)
paraffin)
paraffin)
1
2.3
3.2
0.036
2
7.1
9.7
0.11
5
34
47
0.52
10
115
160
1.8
15
238
330
3.7
mate ranges of fast protons in standard air and in paraffin, a typical
radiator material. It may be seen that, for neutrons of energy ~1 Mev
420 The Neutron [Pt. VII
or less, "thin" radiators must be very thin indeed. As the radiator thick
ness becomes comparable to the proton range, the energy resolution of
the detector rapidly deteriorates. For thick radiators (thickness >
maximum proton recoil range) the situation again arises wherein a
monoenergetic neutron produces pulses of all energies up to its own.
However, the pulse height distribution is much more complicated than
that from neutrons scattered in the chamber gas. For neutrons of
energy E n incident normally on a thick radiator, with the chamber
counting all pulses above the bias energy E t , the sensitivity function is
approximately (B9)
S(E n ,E t ) g* ke(E n )E n  y \E n y>  E t %A f (110a")
in which the rangeenergy relationship has been approximated by
B(E P ) oc E P 3A . This function has zero slope at E n = E t , and is a mono
tonically increasing function of E n ; hence, such a chamber is most sensi
tive to the highestenergy neutrons in the beam.
(e) Other Possibilities. In principle, any neutron reaction can be used
as the basis for a neutron detector. Thus, the radiative capture process
can be observed by detection of the prompt capture gammarays; in
elastic scattering can be detected by observation of the accompanying
gammaradiation. With the recent development of scintillation counter
techniques, the efficiency of gammaray detection is comparable with
that for the detection of charged particles.
One application of these techniques to neutron counting is of special
interest in that it can be used to provide an efficient detector of inter
mediateenergy neutrons (D16). This technique takes advantage of the
fact that, for slow and intermediate neutrons, the B 10 (w,a) reaction goes
mainly to the 0.48Mev excited state of Li 7 , from whose decay the
gammaray can be detected by a scintillation counter. Since, in such a
detector, solid boron can be used, the size being limited only by the
penetrability of the 0.48Mev gammaray, high efficiencies are easily
obtainable.
As previously pointed out, charged particle reactions are easily ob
servable if they take place in a scintillating material. Thus, Hofstadter
et al. have detected thermal neutrons, with high efficiency, by the scin
tillations produced, in a Lil(Tl) crystal, as a result of the Li 6 (n,a)
reaction (H64). Such reactions can also be observed by allowing the
charged particle to impinge, from the outside, on a scintillator. The
zinc sulfide screen, so important in the pioneer work on natural radio
activity, has been used in this connection (M38).
Other charged particle reactions, both exoergic and endoergic, can be
used for ionization chamber and proportional counter materials. Among
Sec. 3C] Sources and Detectors 421
these, the N 14 (n,p) reaction, Q = 0.626 Mev, has possibilities as an
intermediateneutron detector, since its relatively low Q value permits
identification of the energy of the neutron responsible for the reaction
through a pulse height measurement (F7). Unfortunately, the presence
of resonances in this reaction, for neutron energies above ~500 kev,
limits its usefulness to intermediateenergyneutron spectroscopy.
Other reactions with similar characteristics are He 3 (n,p) and Cl 3S (n,p).
Finally, recoils from elastic scattering by light nuclei other than H 1
can be used for neutron detection. In this regard, He 4 recoils have the
advantage of shorter range (thereby decreasing the wall effect correc
tion) and less energy spread for a given incident neutron energy.
4. Photographic Emulsion and Cloud Chamber Detectors. Photo
graphic emulsions and cloud chambers have been among the most use
ful tools for the observation of nuclear reactions. Both of these devices
have the property that the path traversed by a charged particle appears
as a visible track. From the length of the track (range) and the density
of ionization (grain or droplet density), it is possible to determine the
energy and velocity, respectively, of the ionizing particle. A curve of
ionization density vs. residual range for a track uniquely determines the
energy, mass, and magnitude of charge of the particle. Measurement
of the multiple Coulomb scattering in the photographic emulsion or in
the gas or plates in a cloud chamber provides a measure of the particle
energy. The curvature of the track in a known magnetic field yields the
particle's momentum, as well as the sign of its charge. Since track
curvature results both from smallangle Coulomb scattering and from
applied magnetic fields, the use of magnetic fields is usually confined to
cloud chambers, or to regions between two photographic emulsions,
since the Coulomb scattering is smaller and the range is greater in a gas
than in the emulsion; however, for sufficiently strong fields and particles
of high energy, curvature due to magnetic fields has been observed on
tracks in emulsions (D14).
In essence, photographic emulsions and cloud chambers yield the same
sort of information; indeed, the photographic emulsion can be said to be
identical with a cloud chamber operated at very high pressure. How
ever, owing to the differences in their mode of operation, emulsions and
cloud chambers should be regarded as complementary instruments, the
choice between the two being primarily determined by the nature of the
problem under investigation. Cloud chambers have the advantage that
it is easier to choose and change the filling gas. On the other hand,
emulsions are continuously sensitive, and are therefore much more use
ful in experiments involving weak sources. Furthermore, the sensitivity
of emulsions is variable over a very wide range — emulsions are available
422 The Neutron [Pt. VII
which are insensitive to tracks of ionization density less than that of a
fission fragment; at the other extreme, it is possible to obtain emulsions
which show tracks of electrons at the minimum of the ionization rate
curve. Another advantage of emulsions is their small size and light
weight.
The main disadvantage in the use of photographic emulsions arises
from the short range, in them, of charged particles. (A 1Mev proton
has a range of ~15 /*, a 1Mev alphaparticle of ~3.5 /j..) As a result,
it is necessary to detect and measure tracks with a microscope, and the
accumulation of large amounts of data becomes a relatively tedious
affair compared to experiments involving electronic detection. How
ever, the wide flexibility and the possibility of unambiguous identifica
tion of reactions frequently more than compensate for this disadvantage.
The earliest experiments with photographic emulsion detection of
charged particles employed xray and "halftone" plates. Thus, Taylor
and Goldhaber (T4), as early as 1935, used boronimpregnated Ilford
R emulsions, irradiated by slow neutrons, to prove that the B(n,a) reac
tion involves the emission of a single alphaparticle. However, the
emulsions available at that time could only detect lowenergy charged
particles; their lack of sensitivity and uniformity greatly discouraged
their use in nuclear physics. Since World War II, tremendous progress
has been made in the development of more sensitive emulsions and of
techniques for their use, in large measure through the instigation and
inspiration of Occhialini, Powell, and their collaborators. Emulsions
called "nuclear emulsions," with various degrees of sensitivity, and with
various thicknesses, from 25 to 600 fi, are now available. (Thicker emul
sions can be obtained by special arrangement with the manufacturers.)
The use of nuclear emulsions has been extensively described by Powell
and Occhialini (P25), by Yagoda (Y2), and in a recent summary by
Rotblat (R22). Rotblat also gives the most recent data on the composi
tion and properties of the various available emulsions (his Tables I, II,
and III) and on processing techniques.
There is an extensive literature on cloud chambers. A recent mono
graph by Wilson (W34) summarizes the techniques of cloud chamber
operation. Further details on these subjects are given in Volume I,
Part I.
(a) Detection of Slow Neutrons. Nuclear emulsions contain an appre
ciable amount of nitrogen (0.073, 0.080, 0.11 g/cm 3 , respectively, in Il
ford, Kodak, and Eastman Kodak emulsions). The N 14 (n,p) reaction
has a Q value of 0.626 Mev, with a cross section a' = 1.76 barns for
thermal neutrons. The protons resulting from thermal neutron capture
Sec. 3C] Sources and Detectors 423
(range ~7 n) can be observed and counted in the emulsion, thereby
providing a measure of the thermal neutron flux (C33).
The sensitivity of nuclear emulsions to slow neutrons can be greatly
enhanced by adding small amounts (plates are available with ~1% by
weight) of lithium or boron to the emulsions. The (n,a) reactions in
such loaded emulsions can be used for measuring even small slowneutron
fluxes with relatively high accuracy (S25, T10, B35, R22).
It is also possible to impregnate emulsions with uranium acetate or
citrate, and to observe tracks due to fission by slow neutrons (M43).
Owing to the high background of alphaparticle tracks from the uranium,
such plates must ordinarily be exposed and processed within a short
time after preparation. However, emulsions are available whose sensi
tivity to alphaparticles has been greatly decreased, so that only the
fission fragment tracks are observable (Ilford Dl, Eastman Kodak
NTC).
It is also possible to measure slowneutron fluxes by observing the
blackening of xray and electron sensitive emulsions resulting from slow
neutroninduced radioactivity (B35).
The reactions mentioned above can also be observed in cloud cham
bers, although it is considerably more difficult to obtain a suitable gas
containing lithium, boron, or uranium; instead, these elements are
usually introduced into the chamber as thin coatings on plates.
(b) The Observation of Proton Recoils. Nuclear emulsions contain an
appreciable hydrogen content (~0.040.05 g/cm 3 in dry emulsions;
considerably more in a moist atmosphere). Thus, irradiation with fast
neutrons will give rise to proton recoil tracks.
Because of the variation of the proton recoil energy with angle, accord
ing to Eq. (108), measurement of the energy of a proton recoil will yield
the neutron energy only if the relative directions of neutron and proton
are known. In other words, to use proton recoils for neutron energy
and flux measurements, it is necessary for the incident neutrons to have
a welldefined direction. (The previous discussion of proton recoil
ionization chambers and proportional counters is fully applicable to this
section.) Thus, in measuring a fastneutron flux distribution by ob
serving the proton recoil range spectrum, it is necessary to set up strict
criteria for the acceptance of tracks. Such criteria usually involve the
choice of a limiting angle, O of Eqs. (109") and (108a). In observing
tracks which "dip" in the emulsion, it must be kept in mind that emul
sions shrink by a factor ~2 in being processed.
It is relatively easy to make corrections for the background due to
other (n,p) reactions in the emulsion — say on N 14 — or due to scattered
neutrons which strike the emulsion from all directions. Since the recoil
424
The Neutron
[Pt. VII
protons from neutrons in the beam are confined to the forward hemi
sphere and the background protons are usually distributed with approxi
mate spherical symmetry, measurement of the proton recoil spectrum
in the backward hemisphere can provide the necessary data for this
correction.
For neutrons of energy greater than ~1Q Mev, an appreciable frac
tion of the recoil protons leave the emulsion before coming to the end of
their ranges; the range of a 10Mev proton is ~600 n of emulsion. Thus,
as the neutron energy is increased, an appreciable correction must be
applied for the loss of tracks. This correction can be decreased by using
0.3
^.0.2
Li
.
\
a a
J>
^ s
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
•E„(Mev)
Fig. 49. Li 6 (n,a)H s cross section for normal lithium (7.4%Li 6 , 92.6%Li 7 ) due to
J. M. Blair et al. (unpublished) (G20, A2).
thicker emulsions. For very fast and especially ultrafast neutrons, it is
more accurate to measure the proton energy by the track's grain density,
or by the smallangle scattering.
In the intermediateenergy range, the proton recoil method is limited
by the short range of the recoil protons, which renders inaccurate the
measurement of range and of angle. Most experimenters regard 1 Mev
as a practical lower limit of the neutron energies to which the technique
is applicable. Recently, Nereson and Reines (Nl) have shown that, by
careful application of the technique, accurate neutron flux measure
ments can be made down to E n = 0.5 Mev, and possibly pushed (with
poor resolution and large corrections) to 0.3 Mev.
The use of hydrogencontaining cloud chambers for neutron detec
tion is subject to the same considerations as have been outlined above.
(c) Other Reactions — Especially Li 6 (n,a)H 3 . For the detection of fast
neutrons by a charged particle reaction in nuclear emulsions, it is neces
sary that the cross section for the reaction in the emulsion be at least
comparable with that of neutronproton scattering, and that the reaction
products be distinguishable from the recoil protons. In the energy range
~0.110 Mev, the Li 6 (n,a) reaction, Q = 4.79 Mev, has the desired
Sec. 3C]
Sources and Detectors
425
properties. The (n,a) cross section of normal lithium (7.4% Li 6 ) is
shown in Fig. 49 for neutron energies up to 0.8 Mev.
In order to overcome the large proton recoil background, it is neces
sary to employ emulsions impregnated with enriched Li 6 . Such emul
sions have been used by Keepin and Roberts (K5) (Eastman Kodak
">B*' ,1
' > ''V . I
I. 7 •■■
\
**■• "' \
•09 %
\
t
Wmi§m
$P$$l)!0&i>'
E
>V:
Fig. 50. Photomicrograph of a Li 6 (n,a)He 3 disintegration in a Li 6 loaded NTA
emulsion. The lower track is due to the alphaparticle. The background tracks
are mainly due to proton recoils. Courtesy J. H. Roberts (private communication).
NTA, with 0.04 g/cm 3 Li 6 loading), who have successfully demonstrated
the usefulness of this method of fastneutron detection and flux measure
ment.
In the measurements of Keepin and Roberts, the range of the alpha
particle and of the triton are measured, as well as the angle between
them. They have found that the best energy resolution, ±0.1 Mev for
neutrons from to a few Mev, is obtained by selecting only those events
for which the angle between the particles is between 175° and 180°. It
is necessary to use special processing techniques to permit discrimination
between the alphaparticle and triton tracks. A photomicrograph of a
typical event is shown in Fig. 50.
426 The Neutron [Pt. VII
The abovementioned resolution of ±0.1 Mev has been achieved for
collimated neutron sources. With isotropic neutron fluxes, for which
this method is possible where the proton recoil method cannot be used,
such good energy resolution has not been attained (R12).
For neutrons of energy greater than ~5 Mev, the very large back
ground of protons of comparable range renders the use of the Li 6 (w,o:)
reaction relatively inaccurate. However, for E n > 10 Mev, the
B 10 (n,2a)H 3 reaction has been successfully used for neutron detection
with boronimpregnated emulsions (L5).
D. SlowNeutron Spectroscopy
Probably more work has been done, and more information obtained,
on the properties of neutrons between and 10 ev energy than on all
the rest of the neutron energy spectrum. This was certainly true before
and during World War II, and continues, to a somewhat lesser extent,
to be true today. This emphasis has not been entirely due to the greater
ease of producing and detecting slow neutrons. The existence and the
properties of distinct, narrow, slowneutron resonances provide a con
firmation and a test of the ideas of Bohr and his followers on the proper
ties of the compound nucleus, and on the possibility of a statistical
treatment of nuclei. Furthermore, the wave properties of the neutron
and the fact that slowneutron wavelengths are of the order of inter
atomic distances in molecules, liquids, and solids hold forth the intriguing
prospect of using slow neutrons as a tool for the study of the structure of
matter. While a number of such studies were carried out in pre World
War II days, this aspect of slowneutron physics is only now, owing to
the availability of the strong sources from piles, bearing fruit; these
studies will be the subject of Section 5.
The history of the development of techniques for slowneutron investi
gations was summarized in Section 2A. In this section these techniques
are discussed in detail. Although many types of experiments can be
designed to measure the variation of some effect or other with neutron
energy, the quantity most frequently measured is the cross section.
The cross section is usually measured by a transmission experiment.
A "good geometry" transmission experiment measures the total cross
section, <r = £) cr,, the sum of the cross sections for all processes which
i
remove neutrons from a beam; these include scattering as well as ab
sorption processes. The quantity measured in a transmission experi
ment is the attenuation
y = e~ N ° T (112)
Sec. 3D] Sources and Detectors 427
the ratio of neutron intensities at the detector with and without an
absorber (NT = number of absorber atoms per square centimeter of
absorber *) in the beam. Such a measurement does not require any
knowledge of the value of the neutron flux.
In order to obtain maximum accuracy in a minimum of counting
time, especially if the available beam intensities are not very great, it is
necessary to make a proper choice of the absorber thickness. The opti
mum thickness, which depends on the details of the experimental
arrangement and on the background, is, over a wide range of experi
mental conditions, for NaT « 12 (R17, R9).
Since scattering is always one of the components of the total cross
section, it is necessary, in a good geometry experiment, to correct the
measured transmission for those scattering processes in which the de
flected neutron strikes the detector. 2 The smallness of the correction is
the measure of the goodness of the geometry. In order to make this
correction, it is necessary to know the magnitude and the angular dis
tribution of the scattering cross section.
Under certain circumstances, a "poor geometry" transmission experi
ment can yield independent and useful information. Consider a beam
of neutrons and an absorber, both large in crosssectional area compared
to the detector. Let the detector be placed directly behind the absorber.
The attenuation, I/Iq, due to the absorber is, in this case, e~ N °' T , where
a' = <r — ^<x sc (provided that the scattering is symmetrical with respect
to 90°) since half of the scattered neutrons leave the absorber through
the back face. In all these considerations we assume that the neutrons
have a negligible probability of making more than one nuclear encounter
in the absorber; otherwise, the problem approaches one of diffusion, to
be considered in Section 4.
A still more extreme case of poor geometry is that of a point source
completely surrounded by a (spherical) absorber. In this case, if the
detector is sufficiently far from the source — so that the distribution of
scattered neutrons also resembles that of a point source — the meas
ured cross section is a a = <r — <r sc . Thus, the results of good and poor
geometry transmission experiments enable the determination of both
o sc and <r = <j — a 8c . Experiments in which the geometry is interme
diate between good and poor are rather more difficult to interpret (S65).
1 If the absorber contains more than one type of nucleus, the exponent of Eq.
(112) is replaced by NaT = T £ AW.
k
2 With respect to this correction, inelastic scattering must be regarded as an ab
sorption process if the detector is insensitive to the inelastically scattered neutrons,
and as a scattering process if the detector is sensitive to them.
428 The Neutron [Pt. VII
Transmission experiments are, of course, not the only means of meas
uring neutron cross sections. All the techniques whereby specific reac
tions can be used for measuring neutron fluxes (discussed in Section 3C)
can be reversed, and cross sections determined if the neutron flux is
known. Mention must also be made of the techniques developed by
Dempster and coworkers (L4) and by Inghram, Hess, and Hayden
(II, H50, H55) for the detection of (strong) neutron absorption and
measurement of cross sections by massspectrographic observations of
the changes in isotopic abundances resulting from neutron bombard
ment. The reader's attention is especially called to a beautiful photo
graph published by Dempster [Phys. Rev., 71, 829 (1947)] showing the
decrease in abundance of the Cd 113 isotope when normal cadmium is
subjected to intense neutron irradiation. Finally, a transmission meas
urement can be changed into a direct measurement of <j sc by moving
the detector out of the neutron beam; however, such a measurement
requires either an exact knowledge of the geometry and the neutron
flux, or calibration with a scatterer of known cross section, say carbon
(H38, H40).
1. SlowNeutron Monochromators. Devices for singling out neutrons
of a specific (usually variable) energy are known as monochromators or
velocity selectors. A number of such devices, based on various princi
ples, have been designed and constructed. They all employ, as the
slowneutron source, neutrons emerging from a mass of slowingdown
material (paraffin, heavy water, graphite) whose energy distribution is
approximately Maxwellian, with an epithermal tail (1/E flux distribu
tion) extending into the intermediateenergy range. As neutron detec
tor, a boroncontaining proportional counter is usually employed. For
maximum sensitivity over the widest possible range, the counter is filled
with enriched B 10 F 3 .
Although the type of monochromator employed depends primarily on
the nature of the primary neutron source, there are two basic factors
which determine the usefulness of a given device: (1) the energy range
over which it can be operated, and (2) the resolution at a given energy,
which is a measure of the energy spread at the energy under considera
tion. In order to define the resolution, it is necessary to specify the
shape of the resolution function. The resolution function can usually
be approximated by an isosceles triangle. Unless otherwise specified, a
triangular resolution function will be assumed, the monochromator
energy being defined as the energy at the apex, and the "resolution" as
the energy spread at the base. The characteristics of some of the best
existing monochromators are summarized in Table 29. The figures in
the third column for the "time of flight" velocity selectors give the ratio
Sec. 3D] Sources and Detectors 429
of the time resolution of the instrument (in microseconds) to the flight
path (in meters) ; this ratio sets the basic limitation both on the usable
energy range and on the resolution of the velocity selector. The last
five columns give the energy resolutions which are normally attained in
operation.
TABLE 29
Characteristics of Some SlowNeutron Monochromators
Usable Range
(ev)
Resolution
0xsec/m)
Resolution (ev)
0.025
0.1
1
10
100
1,000
Argonne thermal chopper
(B65)
Argonne fast chopper (S18,
S19)
Columbia modulated cyclo
tron (R4)
Harwell electron accelerator
(M26)
Crystal spectrometer (S56)
0.0020.2
0.001 5,000
0.00110,000
0.00120,000
0.02 100
(50)
0.6
1
0.5
0.005
0.001
0.001
0.001
0.0015
0.05
0.05
0.05
0.05
0.01
0.1
0.07
0.1
0.2
1.7
1.2
1.3
5
40
30
20
1,000
1,000
500
(a) Mechanical Velocity Selectors or Choppers. The first slowneutron
monochromator, built by Dunning, Pegram, Fink, Mitchell, and Segre
(D20), was based on the principle used by Fizeau in his classic measure
ments of the velocity of light. In this device two cadmium disks were
mounted, one at each end, on a shaft. Both disks had a series of uni
formly spaced radial slits; the disks could be displaced by an arbitrary
angle with respect to each other. With the disks so misaligned, thermal
neutrons, moving along the direction of the shaft, which passed through
the first set of slits were absorbed in the second disk, provided that the
shaft remained stationary. However, with the shaft rotating, those
neutrons which passed through the slits of the first disk, whose time of
flight between disks was equal to the time required for the second disk
to move through the angle of misalignment, could also pass through the
second slits and be detected. Thus, by varying the velocity of rotation
or the angular displacement, it was possible to select neutrons of a pre
determined velocity. Although the resolution of this device was poor
(Sl3a), the velocity selector was useful in demonstrating the 1/v nature
of a number of thermal neutron cross sections. Improvements and
further measurements on this velocity selector were carried out by Fink
(F29).
Taking advantage of the greater thermal neutron fluxes available
from the first pile, Fermi, Marshall, and Marshall (F22) constructed a
430
The Neutron
[Pt. VII
thermal neutron velocity selector based on the same principle, but with
a number of significant modifications. In this device a beam of neutrons,
from the thermal column, falls on a cylinder whose axis is perpendicular
to the beam direction. The cylinder is made up of alternate, lengthwise,
thin layers of aluminum (effectively transparent to thermal neutrons)
and cadmium, as shown in Fig. 51. Neutrons can pass through the
cylinder only when the layers are parallel, within ±3°, to the beam
direction. Thus, when the disk is rotated (at speeds up to 15,000 rpm)
about its axis, neutrons are transmitted in short bursts.
Aluminum v\^
Steel
W
Cadmium 1
Multiple sandwich
of 0.004" to 0.008"
cadmium and
aluminum foils
Fig. 51. Cross section, perpendicular to the axis of rotation, of the mechanical
"chopper" used in the thermal neutron velocity selector of Fermi, Marshall, and
Marshall (F22, B65).
The neutrons which get through the shutter are detected in a bank of
BF3 proportional counters, 1.46 meters away. The neutron time of
flight between shutter and detector is determined electronically, by
allowing the neutron detector to be sensitive only for short periods,
delayed by an arbitrary time interval t with respect to the shutteropen
times. The synchronization between the shutter position and the de
tectoron time is achieved by attaching to the cylinder axis a mirror
which reflects a beam of light into a photocell when the shutter has a
given orientation. By varying the position of the photocell, the delay
time t can be chosen at will. Only those neutrons will be recorded whose
time of flight between the shutter and the detector is equal to t. If the
burst time duration and detectoron times are both equal to At, and
both have square shapes, the velocity resolution function is a triangle
with apex at the velocity v = L/t and base width Av = 2v At/t =
2v 2 At/L, where L = the length of the flight path.
Sec. 3D]
Sources and Detectors
431
This velocity selector was subsequently improved by Brill and Lich
tenberger (B65), who used it to measure the cross sections of a number
of elements in the thermal neutron region. The numbers given in the
first line of Table 29 apply to the improved model.
Velocity selectors which employ cadmium in the neutron shutter
cannot be used for energies above the cadmium cutoff. This limitation
Pig. 52a. The fast chopper of Selove, at the Argonne Laboratory (S18, S19). Pho
tograph shows the stator, rotor, and photocell (box) for timing.
does not apply to the "fast chopper" constructed by Selove at the Ar
gonne laboratory (S18, S19), which employs, as a neutron shutter, a
long cylinder of steel with thin slits cut into the cylinder parallel to the
axis, which is also the direction of the neutron beam. In line with this
cylinder, which can be rotated about its axis, there is a stationary cylin
der with identical slits. Neutrons can pass through both sets of slits
only when the two sets are aligned; when the slit systems are misaligned,
the thickness of steel is sufficient to remove neutrons of all energies from
the beam. This device is shown in Figs. 52a and 52b.
The neutron detection and timing system is, in principle, the same as
that of the thermal chopper. In operation, the flight path L = 10
meters and the burst length A* « 6 n sec. The characteristics of this
velocity selector are given in the second line of Table 29. The possibil
ity of using such a relatively high resolution device is dependent upon
432
The Neutron
[Pt. VII
Slit width (W)
Fig. 52b. Details of the rotor construction of the chopper in Fig. 52a.
1.0
0.5
"f!'^>
Y\
J ^A { •
T\ l?~c
r ■• "\o/^~7
W 182
a i.o
0.5
1.0
0.5
10
20
IX sec/m
30
g/cm 2
w 182 W 183 w 184 w 186
1.58
0.054 0.07 0.021
1
o
°„
N^
>*~~«»
VS.
o °
V
W 186
007 0.006 0.023
X ^\
AH
^ * ,
f\!
•\ L><
\
1 '
fv/
\J
Normal
wolfram
8.04E/cm 2 242 240
(3Q.H)(29.8»)
(22JJ) j 39
U73J)
5
5000 500
1 1 1 i — i — i — i — i — i 1 — i — i — i — r
1000 200 100 50 30 20 15 10 8 6 5 4
Fig. 52c. Transmission curves for separated wolfram (W) isotopes, made with the
fast chopper of Fig. 52a.
Sec. 3D] Sources and Detectors 441
lowenergy side of the resonance never falls below 2000 barns. On the
highenergy side, on the other hand, the cross section descends abruptly;
above ~4 ev the cross section (5 barns) is due entirely to scattering'
there being no additional resonances observed up to ~100 ev. A thick
ness of 1 g/cm 2 (~l mm) of cadmium metal will transmit less than 1
percent of all the incident neutrons of energy below 0.3 ev, and ~M of
the neutrons of energy 0.45 ev.
Only a negligible portion of the Maxwell distribution (for reasonable
neutron temperatures) extends into the epicadmium region. On the
other hand, if the absorber has a resonance fairly close to the thermal
region, the lowenergy tail of the resonance may extend to below the
cadmium cutoff; in this case, the absorption of a cadmiumcovered de
tector will be less than the true resonance absorption, and it is neces
sary to apply a correction (increase) to the cadmiumcovered detector
reading before subtracting it from the bare detector value, in order to
obtain the true thermal neutron absorption. The magnitude of this
correction will depend on the thickness of the cadmium shield.
Indium is frequently used for a thermal neutron detector because of
the magnitude of its cross section (<J ^ 190 barns), the ease of handling
indium metal foils, and the convenient halflife of the resulting radioac
tivity (54 min). The lowest indium resonance is at 1.44 ev (<r ^ 35,000
barns, T s 0.08 ev; see Fig. 7c, page 281). The correction to the activ
ity of cadmiumcovered indium foils to obtain the true resonance activa
tion has been investigated experimentally by Kunstadter (K26), who
gives the following correction formula:
True activity = observed activity X e 0A3ST (115)
where T is the thickness of cadmium metal in millimeters. 1
(6) The Pile "Danger Coefficient" Method. A useful and accurate
method of measuring thermal neutron absorption cross sections has been
reported by Anderson, Fermi, Wattenberg, Weil, and Zinn (A25). This
method involves the use of a thermal pile (see Section 4), and depends
on the fact that introduction of absorbing material into the pile causes
a decrease in the pile reactivity. If the pile is operating at a constant
neutron flux level, the insertion of the absorber will result in a gradual
decrease in the flux; the effect of the absorber can be compensated by
displacing the pile control rods by an amount sufficient to maintain the
pile at a constant power level.
If the compensating motion of the control rod is calibrated against a
series of absorbers of known cross section, the required motion of the
1 Below ~0.5 mm of cadmium, the shield is no longer black to thermal neutrons,
and the above considerations break down.
442 The Neutron [Pt. VII
control rod for an unknown absorber provides a measure of the un
known absorption cross section. This method was first devised in order
to ascertain the harmful effects (with respect to pile reactivity) of various
contemplated pile construction materials; hence the term danger co
efficient.
One of the characteristics of this method is that it measures the
absorption cross section, in contrast to the total (absorption plus scat
tering) and the activation (absorption leading to a measurable radio
activity) cross sections. The pile reactivity is relatively insensitive to
the scattering properties of the absorber, even if the absorption cross
section is comparatively small. Since the products of the particular
absorbing reaction (provided they are not neutrons) are of no concern
to the pile, this method provides one of the few means of observing ab
sorptions which lead to a stable product nucleus, or to a product nucleus
whose radioactivity is difficult to observe; for example, although activa
tion measurements (S24) seemed, at first, to yield an apparent absorp
tion cross section of 0.02 barn for niobium, which has the single isotope
4iNb 03 , the danger coefficient method gives a value of 1.4 barns (A25).
The discrepancy was resolved when a previously unrecognized isomer of
Nb 94 was discovered.
A modification of the danger coefficient method suggested by Wigner
greatly increases its sensitivity. This is the method of the "pile oscilla
tor," in which the absorber is intermittently introduced into the pile,
resulting in an oscillation of the flux (power) level of the pile. The
magnitude of the resulting pile oscillation is a measure of the absorption
cross section of the sample. The effect of scattering by the sample is to
introduce an outof phase component into the pile oscillation; accord
ingly, the effects of scattering and absorption can be separated experi
mentally, and the absorption of samples for which the absorption cross
section is only a small fraction of the scattering cross section (e.g., bis
muth) can be measured.
The theory of the pile oscillator has been developed by Weinberg and
Schweinler (W12). Extensive pile oscillator measurements of thermal
neutron absorption cross sections have been made at the Argonne
Laboratory by Harris, Muehlhause, Rasmussen, Schroeder, and Thomas
(H42), who used an oscillator constructed by Langsdorf (L3), and at
the Oak Ridge National Laboratory by Pomerance (P22, P23) with the
oscillator of Hoover, Jordan, Moak, Pardue, Pomerance, Strong, and
Wollan (H65). A pile oscillator has also been developed and used with
the French heavy water reactor by Raievski and Yvon (R2).
In using a thermal pile for a danger coefficient measurement of a
thermal neutron absorption cross section, it is necessary to correct for
Sec. 3D] Sources and Detectors 443
the absorption of an appreciable epithermal neutron component. This
correction can be determined by the cadmium difference method.
The danger coefficient method can also be applied to the measurement
of epithermal absorption cross sections (cadmiumcovered foils) (L3).
It can be used with intermediate and fastneutron piles. In these cases
the absorption is an average over the relatively broad neutron flux dis
tribution in the pile. However, in these applications it is much more
difficult to correct for the effects of the scattering properties of the
sample.
3. Measurement of the Characteristics of SlowNeutron Resonances.
One of the most important problems of slowneutron spectroscopy is the
precise determination of the constants associated with resonances. In
the following discussion, we consider only capture resonances, neglecting
scattering and associated interference effects; i.e., we assume r « r r
J£> r„. (Similar considerations can be applied, with minor modifications,
to scattering resonances.) In this case, the BreitWigner formula is
most conveniently written
(E r /E)%
° m  m^wn <n6)
A resonance is completely described in terms of three parameters: E r ,
the resonance energy; o , the peak cross section; T, the full width at
half maximum.
Equation (116) is not symmetrical about the energy E r , owing to the
factor (Er/E)^. The variation of this factor is important for resonances
occurring in or near the thermal region. However, for resonances for
which T « E r , the deviation of the factor (E r /E) 1A from 1 is negligible
over the region of significant values of v(E) ; this situation prevails for
practically all the observed resonances for which E r ^ 1 ev.
(a) Monochromator Measurements. In principle, a measurement of
u{E) vs. E provides all possible resonance data. In practice, the finite
resolution of the monochromator has a profound influence on the ob
served shape of the resonance, and on the possibility of deducing the
resonance parameters. In the following considerations of the effects
of finite resolution we assume a rectangular resolution function, of
width R, mainly for ease of computation. The computations can be
performed for triangular, or any other shape, resolution functions by
numerical iteration if necessary; furthermore, a resolution function of
arbitrary shape can usually be approximated with reasonable accuracy
by the superposition of a number of rectangles.
Since monoehromators measure <r(E) by transmission experiments, the
absorber thickness is important to these considerations. We define (1)
444 The Neutron [Pt. VII
a thin absorber as one which is at least translucent over the entire
resonance, i.e., NT<r < 3; (2) a thick absorber is black in the region of
the resonance, NTcr > 10. It should be remarked that the choice be
tween using a thin or a thick absorber is not entirely a question of the
available techniques of absorber preparation. The monochromator
resolution is the determining factor: for R » T it is impossible to obtain
sufficient absorption, and consequently to make any measurement at
all, by using a thin absorber. On the other hand, for R < r the use of
thin absorbers is, as we shall see, possible and desirable.
The available monochromator resolutions have been summarized in
Table 29. Since resonance widths are, for heavy nuclei, ~0.1 ev, good
resolution (R < T) is possible only below ~1 ev; in the region 110 ev,
the available resolutions are fair; above ~10 ev, resolutions are poor.
The situation with respect to scattering resonances in medium nuclei is
somewhat better; however, very few such resonances fall in the slow
neutron region. The considerations of this section can also be applied
to measurements in the intermediate and fast regions, where many scat
tering resonances have been observed.
(1) Thin Absorber Measurements. For a sufficiently thin absorber,
the transmission is determined by the average, over the resolution func
tion, of the cross section. Neglecting the variation of (E r /E) V2 ,
.E+RI2
&(E) =  I o{E) dE
R Je—R/2
itan
2R
2(E  E r ) R
r rj
— tan 1
2(E  E r ) R\
r rl
(116a)
Curves of a/cr are plotted in Fig. 56 as a function of 2(2? — E r )/T, for a
number of values of R/T. For R/T < 1, the main effect of the finite
resolution is to depress the peak without greatly broadening the reso
nance (although the observed width at halfmaximum is somewhat in
creased). Thus, for good resolution experiments, it is relatively easy to
make the appropriate corrections to the transmission curves, and to
determine the resonance parameters. (The work of McDaniel, Mil,
on the 1.44 ev resonance of indium, see Fig. 7d, illustrates the method.)
However, for R/T > 1 the observed width is essentially the resolu
tion. It is impossible, from such a measurement, to determine sepa
rately the resonance width or the peak cross section; a poor resolution
thin absorber experiment can only yield the product a r, through
Sec. 3D]
Sources and Detectors
445
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446 The Neutron [Pt. VII
either the observed peak cross section or the area under the observed
resonance:
(a) The apparent (observed) peak cross section J is
r _.B
c{E r ) — <r — tan —
R T
R
for  « 1
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r /ir r\ r
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The reduction of the peak cross section (by the factor 1.571 T/R) makes
it virtually impossible to use a thin absorber when R/T > 10.
(b) It may be seen, by visual inspection of Fig. 56, that the decrease
in the peak cross section is accompanied by a raising of the tail of the
observed resonance, so that the area underneath the resonance curve
appears relatively insensitive to the resolution. Actually (from Eqs.
116 and 116a) 2
fff(E) dE = fff(E) dE =  (T r (116b)
independent of R !
Another possible effect of poor resolution is the failure to resolve close
resonances. This difficulty is illustrated in Fig. 57, in which o/oq has
been plotted, for two identical resonances separated by the energy D,
for a number of values of R/T. The classic example of the phenomenon
of close resonances is the case of iodine, which appeared, from early
experiments, to have a single, anomalously broad resonance at ~40 ev.
Careful investigation by Jones (J 12) established the existence of a num
ber of sharp resonances between 20 and 50 ev. Many of the relatively
broad peaks in crosssection curves (G20, A2) observed above ~10 ev
1 Another resolution shape, for which the average can be performed analytically,
is the Gaussian function of halfwidth R. In this case,
HEr) = <r [1.665(r/fi)e o.693rVR 2 erfc (0.833l7#)]
S 1.476<r r/ft for R/T » 1 (116a")
in which erfc (x) = (ir/4)* JJe^dt.
2 The integrals are conventionally and most easily performed between the limits
— x and M. Since the entire contribution is for values relatively close to E r , this
extension of the limits of integration introduces a negligible error.
Sec. 3D] Sources and Detectors 447
in monochromator measurements on heavy nuclei are probably due to
the failure to resolve such groups of resonances.
(2) Thick Absorber Measurements. If a thick absorber is employed
in a good resolution experiment, the transmission is ~0 over the main
portion of the resonance; only the wings of the resonance can be studied
in any detail. However, as we have seen above, thick absorbers are
necessary in a poor resolution experiment to obtain transmissions which
differ significantly from 1.
The interpretation of the transmission curve for a thick absorber in a
poor resolution monochromator experiment can be achieved by numeri
cal and graphical techniques, as. described by Havens and Rainwater
(H45). The result of such interpretation — either by detailed fitting of
the transmission curve or by integration of the absorption over energy
— yields a value of the combination a T 2 . The interpretation requires
a knowledge of the value of the resolution width, R, and of the shape of
the resolution function; the results are, however, not very sensitive to
the exact form of the resolution function.
The fact that the combination a^T 2 determines the results of a poor
resolution thick absorber transmission measurement can be seen by the
following argument: The mean transmission at the energy E is
I/I (E) = ( C R {E' ! E)e' NTn/n+i(  E ' IS ' )Vri] dE'\R(E) (117)
where R(E',E) is the resolution function; the resolution width is de
fined as
R(E) = fR(E',E) dE' (117')
Assuming NT<r > 10, the integrand vanishes unless 4(£" — E T ) 2 /Y 2
^> 1 ; hence, the integral may be replaced by
I/h(E) ^ ( CR(E',E)e NT " oTi/ ^ E ' Er) dE'\R(E)
= F{E,E r ,a Y 2 ) (117a)
The area under the absorption curve is
/'
(1  ///„) dE = F'(E r ,a T 2 ) (117b)
Since almost all monochromator measurements for E ^ 50 ev employ
poor resolution and thick absorbers, a resonance in this region must be
comparatively strong in order to be observed at all. There is no doubt
that such measurements fail to detect a fair fraction of the resonances.
448
The Neutron
[Pt. VII
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Sec. 3D]
Sources and Detectors
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450 The Neutron [Pt. VII
Many of the observed peaks (G20, A2) are probably due to resonance
groups, as in the case of iodine.
Finally, it should be noted that considerable progress has been made
in the direct application of slowneutron monochromators to the study
of the details of resonance scattering, by Tittman, Sheer, Rainwater,
and Havens (T15). The interpretation of measurements on the scat
tered neutrons is quite complicated, but it follows, roughly, the tech
niques outlined above.
(b) Self and MutualIndication Techniques. In the prevelocity
selector era of slowneutron physics, a number of techniques were de
veloped for studying resonances from which it was possible to deduce the
resonance parameters, with surprising accuracy in some cases. While
some of these techniques have been rendered obsolete by more recent
developments, others still find useful application in resonance investiga
tions. Unfortunately, in the first rush to exploit the newfound wealth
of velocity selectors and pile intensities, some of these old methods have
been disparaged, neglected, or forgotten. Their revival should con
tribute useful information to the growing fund of resonance data. These
techniques are admirably summarized in the famous "Bethe Bible"
(B24).
One of these is the method of selfindication, in which the resonance
under investigation is used as its own detector (F16). For a thin reso
nance detector in a slowneutron beam, the saturated activity is (from
Eqs. 103'b, 104, 104a, 103c)
»©
R = NTA<t>(E r ) (  1 «r r (103d)
Thus, knowledge of <r r will directly yield the flux <f>{E r ), and vice versa. 1
If now a thin absorber of the same material (thickness T') is placed
in the beam, the saturated activity of the detector will be reduced,
becoming
R' = NTA f[<j>(E)  NT'^(E)a(E)]o(E) dE
R [l  NT' fa 2 dE/ fa
dE]
= R [L  NT'ao/21 (118)
'We assume, here and in the following, that a = a{n,y); i.e., r ~ l\^i> l'„.
This holds, of course, only for capture resonances in heavy nuclei. The necessary
modifications, if this is not the case, can be obtained by arguments similar to those
employed in this section.
Sec. 3D] Sources and Detectors 451
Thus, the effective cross section for a thin resonance absorber in a self
indication experiment is a /2.
Thus, it is, in principle, possible to measure <r without a monochroma
tor. There are, however, a number of difficulties inherent in the use of
this technique. In the first place, since cr can be very large (~10 4 10 5
barns for resonances at a few ev energy in heavy nuclei), the prepara
tion of thin absorbers may present serious technical problems. The
common solution is to use absorbers of decreasing thickness, and to
extrapolate the resulting values of <r to zero absorber thickness.
Another difficulty arises from the possible activation of the detector
by neutrons of energies outside the resonance. Thermal neutrons are
easily excluded by cadmium. Other resonances are more difficult to
eliminate, especially if they should happen to be of comparable strength
(<r r) to the one in question. Fortunately, this is seldom the case; the
lowest resonance usually dominates the rest so that, especially if ab
sorber and detector are truly thin, the observed effects can be attributed
to a single resonance. It should be noted that the availability of a mono
chromator of even poor resolution would eliminate the effects of other
resonances in a selfindication experiment. However, only the crystal
spectrometer can be used in this connection, since other velocity selec
tors do not remove neutrons from the beam.
When the resonance is used as its own detector, the resonance energy
can be determined by measuring the cross section of boron for the de
tected neutrons. Since the cross section of boron follows the 1/v law,
and its absolute value is known, this method can yield an accurate
value of E r . However, the 1/v dependence of the total (transmission)
boron cross section holds only as long as the absorption cross section is
much greater than the (constant) scattering cross section. This limits
the, use of normal boron to energies below ~100 ev; for absorbers en
riched in B 10 , the range is considerably extended.
An interesting example of the use of these techniques for the study of
a single resonance (rhodium, E r = 1.2 ev) is the work of Manley, Gold
smith, and Sch winger (M3). These authors also used the shape of the
curve of transmission vs. (selfindicated) absorber thickness to obtain a
measure of the resonance width, r. However, as the absorber thickness
is increased, the effects of other resonances are of greater importance.
A natural extension of the technique of selfindication can be made
for the case of two resonances (in different materials) which partially
overlap; one can be used as absorber and the second as detector, thereby
obtaining a measure of the area of overlap of the levels; this is the method
of mutual indication. By measuring all four absorption coefficients (two
452 The Neutron [Pt. VII
self, two mutual) it is possible to obtain the relative widths and reso
nance energies for the two levels (B24, A20, F16, H66).
The techniques described above, with a number of ingenious modifica
tions, have been extensively investigated and exploited by a group at the
University of Groningen, Holland (C27, C28). The technique of
mutual indication has recently been applied by Hibdon and Muehlhause
(H57) to the detection of resonances in a number of substances; they
used the scattering resonances of cobalt and manganese as indicators.
(c) Interpretation of Resonance Data; the Doppler Effect. The fore
going discussion may be summarized as follows: There are available a
number of techniques for the measurement of the constants which char
acterize a single resonance (Eq. 116). Monochromator measurements
determine E r and, for good resolution and thin absorbers, <r and I\
Poor resolution monochromator measurements yield o r for thin ab
sorbers, or cor 2 for thick absorbers. Selfindication measurements
yield o and <r r (if the neutron flux can be measured independently), as
well as E r and, possibly, I\ Mutualindication measurements, in con
junction with another resonance of known characteristics, can give E r
and T.
There is still another method for obtaining information concerning a
resonance, provided that it is permissible to extend the single resonance
formula into the thermal region; this is possible if the resonance in ques
tion is the strongest of all the resonances of the nucleus in question, and
also has the lowest energy. For the thermal energy region, if E th = E
<3C E r , the oneresonance (n,y) cross section is
"^^W (119)
which is the usual 1/v law. Thus, the value of the thermal neutron
capture cross section determines the constant <r T 2 , if E r is known. We
recall that, for a 1/v absorber in a Maxwellian neutron distribution of
temperature T,
(T t h = (95a)
1.128
whence
(t T 2 ^ 1.128cf„, ±E/ A (kTy A (119a)
If this method is used to derive <x T 2 from the thermal neutron capture
cross section of In 115 (Table 10, page 328) the value obtained, ~185
barnsev 2 , agrees, to within ~10 percent, with that computed from the
constants of the 1.44ev resonance as measured by McDaniel (Mil).
Sec. 3D] Sources and Detectors 453
The connection between am and the first resonance depends on the
possibility of neglecting the effect of all higher resonance. The resonance
energy factor in Eq. (119) favors this possibility. The value of or ,
.0 = ^^ (120)
is also a decreasing function, ccE r ~ l/i , of the resonance energy, provided
that the resonances are all of the same type; i.e., T nri ^ CE ri ^ and I\
= constant.
Equation (120) also shows that the neutron width, fT nr , can be ob
tained from a measurement of <r and T (actually, from col 1 ).
In the preceding discussion, r has been associated with the true
width of the resonance. Actually these are not identical, owing to the
Doppler effect of the thermal motion of the absorber nuclei. The Dop
pler effect is equivalent to introducing an inherent finite resolution into
any measurement, since an incident neutron of energy E will have a
spread of energies relative to the absorbing nuclei. The "resolution
function" which describes the Doppler effect is
R D {E',E) = e w*)VA» (121)
where „
A = 2 I J (121a)
and A is the mass number of the absorbing nucleus. The Doppler
width, A, is by no means negligible. For E = 1 ev, kT — 34o ev > A.
= 100, Eq. (121a) gives A = 0.032 ev; A increases like E y \ The effec
tive cross section of a given absorber is
with
, ff (E) = (fa(E')R D (E',E) dE'\R D (E) (121b)
R D (E) = (r d (E',E) dE' = t a A(E) (121c)
Any additional effects due to the finite resolution of the measuring de
vice are superimposed on the Doppler broadening.
The result of this Gaussian resolution function is to reduce the peak
cross section and broaden the resonance in the manner previously de
scribed. 1 The details of the Doppler broadening were derived and dis
cussed by Bethe and Placzek (B25, B24), who give formulas and curves
for the interpretation of the measured constants (also in selfindication
experiments) in terms of the true constants of the resonance.
1 See footnote 1 on page 446.
454 The Neutron [Pt. VII
(d) Effects of Many Resonances; the Resonance Integral. Some medium
and most heavy nuclei have many resonances in the slowneutron region.
The absorption, scattering, and activation of these nuclei in a slow
neutron flux will be due to the combined effect of all the resonances.
Consider a thin detector of a given type of nuclear reaction, with cross
section <ri(E). In the unusual 1/E slowneutron flux, with the thermal
neutrons eliminated by cadmium, the total number of processes of type i
will be (Eqs. 96, 103c", 104b)
Ri = Nvf 4>(E)<Ti(E) dE = NVB f a(E) ( — ) (122)
J~0.5ev J~0.5ev \E/
If <ri(E) is characterized by a number of resonances, their effect is sum
marized by the resonance integral
Recalling that, for a process of type i,
we have
S^2x 2 E 2/j ' r " ryIV (123a)
j TjE r j
Resonance integrals can be evaluated, given a 1/E epicadmium flux,
in a straightforward fashion. The activation of an (n,y) detector gives
2„ for the resonances leading to the observed activity. Pile danger
coefficient measurements can yield S B for all capture processes, includ
ing those which lead to undetectable product nuclei. Measurements of
the total scattering cross section yield S s , provided that a reliable means
can be found for subtracting the effect of the relatively constant poten
tial scattering.
Harris, Muehlhause, and Thomas (H40) have reported measurements
of S a and S s for a number of nuclei. They measured S a by observing
the activation of thin detectors in the 1/E flux next to the Argonne
heavy water reactor. Absolute values were obtained by comparison
with the thermal neutron activation of the same detector, using the
known thermal neutron cross sections and correcting for the effect of
1/v absorption above the cadmium cutoff. The ratio of thermal to
epicadmium flux was determined from the known values of a ath = 93
Sec. 3E] Sources and Detectors 455
barns and S a = 1296 barns (due almost entirely to the resonance at
~5 ev) of gold.
Values of S s were measured by the scattering of thin samples placed
at the center of an annular BF 3 counter. Absolute values were obtained
by comparison with the scattering of a carbon target, for which the
average epicadmium cross section (4.60 barns) was independently deter
mined. Corrections were computed for the (constant) potential scatter
ing, when known. In other cases, the potential scattering contribution
was measured by using a neutron beam which had been filtered through
a thick absorber of the same material as the scatterer (a sort of self
indication in reverse). Unfortunately, the scattering chamber efficiency
depended on the energy of the scattered neutrons, so that the interpreta
tion of the results required a knowledge (or assumption) of the energies
of the most important — i.e., first few — scattering resonances.
The results of these measurements were interpreted in terms of aver
age values of V n /T for the nuclei in question (see Section 2). From the
energy dependence of the factors in Eq. (123a) it is seen that the par
ticular average, deduced from 2 3 /(S a + S s ), is strongly weighted in
favor of the lowestlying resonances.
E. The Calibration of Neutron Sources
Along with the large variety of available neutron sources with dif
ferent spectra and yields, there have been devised a number of methods
for measuring source spectra, some of which have been described in the
preceding, and for measuring source yields — the subject of the following
discussion. Some of these methods are quite specific to the reaction
under investigation. Thus, for charged particle reactions on light nu
clei, the recoil (product) nucleus can frequently be detected and counted
by conventional means, thereby providing a measure of the neutron
yield. An example is the measurement of the yield of the reaction
H 3 (d,n)He 4 by counting the recoil alphaparticles (T2).
Alternatively, if the product nucleus is radioactive, the yield of a
reaction can be obtained from the resulting radioactivity of the target.
This technique requires, however, a determination of the absolute value
of a radioactive decay rate; such measurements are, in general, difficult
to perform with good accuracy. This method is applicable to some of
the threshold (p,n) reactions.
An important variation on this theme is the method of Paneth and
GKickauf (P2, G8) for obtaining the yield of the Be 9 (7,n)Be 8 > 2«
source by measuring the total helium accumulation in the beryllium.
This method could serve as the basis for the calibration of the standard
neutron source (C42).
456 The Neutron [Pt. VII
Another general technique for the measurement of source strengths is
the observation of the excitation of a reaction of known cross section by
the source neutrons. This method requires the use of a detector of
known efficiency and, clearly, is applicable only when the neutrons are
either monoenergetic or have a relatively simple, known spectrum. The
most useful reactions for this application are neutronproton scattering,
for fast neutrons, and the B 10 (n,a) reaction, for intermediate and slow
neutrons.
Most of the useful laboratory sources — radioactive (a,n), photoneu
tron, and many accelerated charged particle reactions — have complex
spectra and are, in general, not susceptible to the abovementioned tech
niques. The calibration of such sources is usually achieved by the
method of "space integration." There are almost as many variations
on this technique as there have been applications thereof. However,
the salient features can be described as follows :
Consider a source of fast or intermediate neutrons in an infinite slow
ingdown medium. 1 Let the source strength be Q (neutrons/sec), the
density of the medium be N (nuclei/cm 3 ), and the absorption cross sec
tion of the medium be a a (E). At equilibrium, we have
Q = 4x I r 2 dr\ N<j>{r,E)<r a {E) dE
ir J r 2 dr I
= 4tN I *(r)o a (r)r 2 dr (124)
where r is the distance from the source. The problem of absolute source
calibration reduces to the experimental determination of the function
*(r)fa(r).
Now, the slowingdown media most commonly used (e.g., water,
paraffin, graphite) have the property that <r a (E) obeys the 1/v law;
there are no known absorption resonances up to at least a few Mev.
Furthermore, the thermal neutron absorption cross sections are small, so
that the absorption during slowing down is negligible; practically all the
absorption takes place after the neutrons have reached thermal equi
librium. Thus, it is possible to measure $(r)o a (r) by using any thermal
neutron detector whose absorption also follows the 1/v law and for
which the ratio cr a (medium)/o (detector) is known. (Since both medium
1 The practical definition of infinite is: large enough so that not more than a small
fraction of the neutrons escape through the outer boundary; actually, if the geom
etry is sufficiently well defined so that it is possible to compute the probability of
escape, this requirement can be relaxed.
Sec. 3E] Sources and Detectors 457
and detector absorb according to the 1/v law, the ratio of their absorp
tion cross sections is independent of the position in the medium.)
As an example consider a small, thin BF 3 counter of known efficiency
(i.e., known N'V). The counting rate of this detector at the position r
in the medium is
R'(r) = N'V'*(rW(r) (103b')
Combining (124) and (103b'), we have
*(£)£)£*<*•*
It is frequently more convenient to use a radioactive (n,y) detector
for the flux measurement. (Manganese and indium foils, using the
cadmiumdifference technique, are convenient 1/v thermal neutron de
tectors.) In this case, a calibration of the detector efficiency is necessary
in order to convert measurements of saturated activity to neutron ab
sorption rates. Such calibrations can be made if the efficiency of the
radioactivity counter is known for the particular foils used. Greater
accuracy of calibration can usually be achieved by comparing the
saturated activity of the foils used with the counting rate of a thin BF 3
counter, of known efficiency, in the same thermal neutron flux; this
method was developed by Frisch, Halban, and Koch (F48).
An interesting technique of foil calibration 1 was carried out by Seidl
and Harris (S16). In this method the absorption rate of a thin boron
absorber, exposed in a thermal neutron flux, is determined by measuring
the accumulation of He due to theB 10 (n,a) reaction. The saturated
activity of the foil used in the integration is measured in the same flux.
This method does not depend on knowing the efficiency of a BF 3 coun
ter. It does, however, require the availability of very strong thermal
neutron fluxes, i.e., a pile.
A useful and ingenious modification of the above techniques has been
described by O'Neal and ScharffGoldhaber (04). Their calibration
method has the advantages that it requires no knowledge of the values
of absorption cross sections and the space integration is performed
physically. The measurement is made in three stages:
(1) The source is placed at the center of a large tank of water in which
is dissolved a quantity of a 1/v detector (they used manganese sulfate) .
After saturation is achieved, the source is removed, the solution
thoroughly stirred, and the activity measured by immersing a Geiger
1 Proposed by L. W. Alvarez.
458 The Neutron [Pt. VII
counter in the solution, at the center. Let the counting rate be i2 x :
Ri = af $i(rKWr 2 dr (125a)
[This method of "physical integration" is described by Anderson, Fermi,
and Szilard (A23.)]
(2) Now mix uniformly into the same solution a known amount
(N'V) of 1/v absorber (they used finely divided manganese powder)
which can be removed from the solution after irradiation. Irradiate to
saturation, remove the absorber, and repeat the activity measurement
on the solution:
R 2 = a] * 2 W^7(r)r 2 dr (125b)
Let R = R 2 /Ri
(3) It is finally necessary to obtain an absolute measurement of the
total saturated activity (rate of neutron absorption) of the absorber:
I = 4irN' I *a(r)«r '(r)r 2 dr (125c)
This step requires the absolute calibration of the radioactivitymeasur
ing device and of the absorber sample used.
Now, since
Q = MiV Hs0 I ®i(r)<TH 2 o(r)r 2 dr
r°"
+ Nunsoi I *iWoMnS0 4 Wr 2 (ir}
•'O
= I + 4n{2V H2 o I *20>h 2 o(?V dr
Jq
+ ^MnS0 4  *2(»MnS04(» 2 cM (125d)
and assuming that all absorption is 1/v, which means
$ 2 W^W = (^) *2«?7« (125e)
in all possible combinations, it follows that
Q = ^ (125)
Sec. 3E] Sources and Detectors 459
A minor modification and inversion of the same technique, first used
by Segre (S14), has been used for the most accurate comparisons of the
absorption cross sections of hydrogen and boron by Whitehouse and
Graham (W19) and others. Consider first a source in a tank of pure
water:
Q = 47riVH 2 o j $o<m^o> 2 dr (126a)
Now dissolve in the tank some boron of density Nb ; then
Q = 47t{^h 2 o' J$i^>r 2 dr + N B f^w 2 dr} (126b)
The activation integral is measured, in each case, with the same 1/v
detector:
I = « I $o<Tar 2 dr
I t = a J Qiw 2 dr (126c)
Since all the absorbers follow the 1/v law, we can apply Eq. (125e),
whence
©■(^) + ©(^) <»>
The difficult step in any of the many methods of source calibration is
the absolute determination, at one point, of the rate of absorption of a
given absorber. Indeed, the various modifications simply shift this
determination from one stage to another. This difficulty need only be
solved once, however, for, after the yield of one source is known, the
yield of any other source can be compared with the standard.
The methods of comparison are exactly those described above, with
the exception that the detector does not require calibration. Thus, the
ratio of the space absorption integrals (Eq. 125c) for two sources in the
same medium, using the same detectors, is the ratio of the source yields.
Actually, if the two sources are of the same type — i.e., same dimensions
and the same energy spectrum — it is merely necessary to compare the
detector activations at one point in the slowingdown medium.
In practice, the use of a resonance detector (e.g., cadmiumshielded
indium) for obtaining the space activation integral is frequently more
convenient, since the curves of resonance activation vs. r fall off more
rapidly than for thermal neutrons. Since the activation of a resonance
detector is proportional to the rate at which neutrons are passing
460 The Neutron [Pt. VII
through the resonance energy (see Section 4) and since, for the usual
moderating media, there is no resonance absorption, the space integral
for a resonance detector is also proportional to the source strength. This
technique derives directly from the early experiments of Amaldi and
Fermi (A20).
An entirely different method of source comparison and calibration
can be used if a pile is available. Consider a pile which is exactly criti
cal, i.e., there is one neutron produced for every neutron lost. If a
source is placed in this pile, the neutron density will increase linearly
with time, since there is no net loss of neutrons in an exactly critical pile.
The rate of rise of the total number of neutrons in the pile, N, is
dN
Tt  Q (127)
In practice, it is necessary to calibrate the pile and neutron detector, and
to correct for the absorption by the source. This method is discussed by
Bretscher (B63).
Alternatively, if the pile is slightly subcritical, and there is a source
in the pile, the equilibrium neutron density will be determined by the
condition that the rate of neutron loss equals the source strength. Since
the rate of neutron loss from a subcritical pile is proportional to the neu
tron density in the pile, the equilibrium density will be proportional to
the source strength, and it can be used (with proper calibration) as a
measure of Q. The theory of this method has been discussed by Placzek
and Volkoff (P18) and by Friedman (F44).
SECTION 4. THE INTERACTION OF NEUTRONS WITH
MATTER IN BULK
A. Introduction
The preceding sections were devoted to the discussion of the interac
tions of neutrons with nuclei and of the instruments and techniques for
the study of neutron phenomena. In this section we consider the suc
cessive interaction of neutrons with nuclei in large masses of material.
For our present purposes the neutrons can be considered to behave in
coherently, their interaction with the medium depending only on the
densities and cross sections of the constituent nuclei. Coherent inter
action phenomena, which depend on the structure of the medium, will
be discussed in Section 5.
The subjects of this chapter are: (1) the diffusion of monoenergetic
neutrons; (2) the slowing down of neutrons; and (3) slowing down and
Sec. 4A] The Interaction of Neutrons with Matter in Bulk 461
diffusion in multiplicative media. The treatment is "elementary"; by
this we mean that simplifying assumptions are made in order to permit
the handling of problems by wellknown mathematical techniques. The
emphasis throughout is on methods of computation which can be used,
with comparative ease, in planning and interpreting experiments. These
methods, although not exact, are sufficiently transparent so as to be
useful for obtaining physical understanding of and insight into the
phenomena under investigation.
The basis principles underlying the mathematic treatments in this
section were laid down by Fermi (F17), soon after the discovery of the
"Fermi effect" — the increase of neutroninduced radioactivity resulting
from the interposing of hydrogenous material between source and detec
tor (F14). In general, the diffusion of neutrons in a medium is govern